Volume 235 No. 1 March 2008 Journal of Mathematics
197
Pacific Journal of Mathematics Volume 235 No. 1 March 2008
Transcript of Volume 235 No. 1 March 2008 Journal of Mathematics
PACIFIC JOURNAL OF MATHEMATICS
Volume 235 No. 1 March 2008
PacificJournalofM
athematics
2008Vol.235,N
o.1
PacificJournal ofMathematics
Volume 235 No. 1 March 2008
PACIFIC JOURNAL OF MATHEMATICS
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University of CaliforniaLos Angeles, CA 90095-1555
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University of CaliforniaSanta Barbara, CA 93106-3080
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The University of Hong KongPokfulam Rd., Hong Kong
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University of CaliforniaLos Angeles, CA 90095-1555
Sorin PopaDepartment of Mathematics
University of CaliforniaLos Angeles, CA 90095-1555
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University of CaliforniaSanta Cruz, CA 95064
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University of CaliforniaLos Angeles, CA 90095-1555
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PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
POWERS OF THETA FUNCTIONS
HENG HUAT CHAN AND SHAUN COOPER
Dedicated to Michael Hirschhorn on the occasion of his sixtieth birthday.
The Ramanujan–Mordell Theorem for sums of an even number of squaresis extended to other quadratic forms. A number of explicit examples isgiven. As an application, the value of the convolution sum∑
1≤m<n/23
σ(m)σ (n − 23m)
is determined, where σ(m) denotes the sum of the divisors of m.
1. Introduction
Throughout this work let τ be a complex number with positive imaginary part, andlet q = e2π iτ . Dedekind’s eta-function is defined by
(1) η(τ)= q1/24∞∏j=1
(1 − q j ).
Let
z = z(τ )=
∞∑m=−∞
∞∑n=−∞
qm2+n2
and 3=3(τ)=η(2τ)12
z6 .
The following result was stated by S. Ramanujan [1916; 2000, p. 159, eq. (14)]and first proved by L. Mordell in [1917].
Theorem 1.1 (Ramanujan–Mordell). Suppose k is a positive integer. Then
zk= Fk(τ )+ zk
∑1≤ j≤(k−1)/4
c j,k3j ,
MSC2000: primary 11E25; secondary 33E05, 11F11, 05A19.Keywords: sum of squares, Ramanujan, convolution sum, modular form, Eisenstein series.The first author is funded by by National University of Singapore Academic Research FundR146000103112.
1
2 HENG HUAT CHAN AND SHAUN COOPER
where c j,k are constants that depend on j and k, and Fk(τ ) is an Eisenstein seriesgiven by:
F1(τ )= 1 + 4∞∑j=1
q j
1 + q2 j = 1 + 4∞∑j=1
(−1) j+1q2 j−1
1 − q2 j−1 ,
and for k ≥ 1,
F2k(τ )= 1 −4k(−1)k
(22k − 1)B2k
∞∑j=1
j2k−1q j
1 − (−1)k+ j q j , and
F2k+1(τ )= 1 +4(−1)k
E2k
∞∑j=1
((2 j)2kq j
1 + q2 j −(−1)k+ j (2 j − 1)2kq2 j−1
1 − q2 j−1
).
Here Bk and Ek are the Bernoulli numbers and Euler numbers, respectively, de-fined by
xex − 1
=
∞∑k=0
Bk
k!xk and sech x =
∞∑k=0
Ek
k!xk .
For the values k = 1, 2, 3 and 4, the condition 1 ≤ j ≤ (k − 1)/4 is empty, andtherefore Theorem 1.1 gives a representation of z, z2, z3 and z4 solely in termsof an Eisenstein series. These are the familiar Lambert series for sums of 2, 4, 6and 8 squares, originally due to C. G. J. Jacobi [1969]. The result for k = 5 wasknown in part to G. Eisenstein (without proof) [1988, p. 501], and stated in full byJ. Liouville (without proof) in [1866]. The result for k = 6 was known in part toLiouville (without proof) in [1860; 1864]. The results for 1 ≤ k ≤ 9 were proved byJ. W. L. Glaisher in a series of papers culminating in [1907]. The general statementof Theorem 1.1 is due to Ramanujan (without proof) [2000, Eqs. (145)–(147)], andthe first proof is due to Mordell in [1917]. Other proofs of Theorem 1.1 have beengiven by R. A. Rankin in [1977, pp. 241–244] and S. Cooper in [2001].
The goal of this work is to prove the analogue of the Ramanujan–Mordell The-orem for which the quadratic form m2
+ n2 in the definition of z is replaced withm2
+mn+n2, m2+mn+2n2, m2
+mn+3n2, m2+mn+6n2, or 2m2
+mn+3n2.Before stating the result we make some definitions. For k ≥1, define the normalizedEisenstein series by
(2) E2k(τ )= 1 −4kB2k
∞∑j=1
j2k−1q j
1 − q j ,
POWERS OF THETA FUNCTIONS 3
where B2k denotes the Bernoulli numbers. Let p be an odd prime. The generalizedBernoulli numbers Bk,p are defined by
(3)x
epx − 1
p−1∑j=1
(jp
)e j x
=
∞∑k=0
Bk,pxk
k!,
where(
·
p
)is the Legendre symbol. Let k be a positive integer which satisfies
k ≡p − 1
2(mod 2).
The generalized Eisenstein series E0k (τ ;χp) and E∞
k (τ ;χp) are defined by
E0k (τ ;χp)= δk,1 −
2kBk,p
∞∑j=1
j k−1
1 − q pj
p−1∑`=1
(`
p
)q j`, and
E∞
k (τ ;χp)= 1 −2k
Bk,p
∞∑j=1
(jp
)j k−1q j
1 − q j ,
where δm,n is the Kronecker delta function, defined by
δm,n =
{1 if m = n,0 if m 6= n.
If p is a prime of the form p ≡ 3 (mod 4), let
(4) F1(τ ; p)= E∞
1 (τ ;χp),
and for k ≥ 1, let
F2k(τ ; p)=E2k(τ )+ (−p)k E2k(pτ)
1 + (−p)k,(5)
F2k+1(τ ; p)= E∞
2k+1(τ ;χp)+ (−p)k E02k+1(τ ;χp).(6)
For p = 3, 7, 11 or 23, let
(7) z p = z p(τ )=
∞∑m=−∞
∞∑n=−∞
qm2+mn+(p+1)n2/4
and
(8) 3p =3p(τ )=
(η(τ)η(pτ)z p
)24/(p+1).
4 HENG HUAT CHAN AND SHAUN COOPER
Furthermore, let
(9) z′
23 = z′
23(τ )=
∞∑m=−∞
∞∑n=−∞
q2m2+mn+3n2
and
(10) 3′
23 =3′
23(τ )=η(τ)η(23τ)
z′
23.
The analogue of the Ramanujan–Mordell Theorem, and the main result of thiswork, is:
Theorem 1.2. Suppose p = 3, 7, 11 or 23 and let k be a positive integer. LetFk(τ ; p), z p and 3p be defined by (4)–(8). Then
zkp = Fk(τ ; p)+ zk
p
∑1≤ j≤(p+1)k/24
cp,k, j3jp,
where cp,k, j are numerical constants that depend only on p, k and j .A similar result holds for z′
23 and 3′
23 defined by (9) and (10), namely
z′
23k= Fk(τ ; 23)+ z′
23k∑
1≤ j≤k
ak, j3′
23j ,
where ak, j are numerical constants that depend only on k and j .
A proof of Theorem 1.2 will be given in Section 2. In the remainder of this sectionwe describe some special cases of Theorem 1.2.
Example 1. For k = 1 and p = 3, 7 or 11, Theorem 1.2 gives
∞∑m=−∞
∞∑n=−∞
qm2+mn+n2
= 1 + 6∞∑j=1
(j3
)q j
1 − q j ,
∞∑m=−∞
∞∑n=−∞
qm2+mn+2n2
= 1 + 2∞∑j=1
(j7
)q j
1 − q j ,
∞∑m=−∞
∞∑n=−∞
qm2+mn+3n2
= 1 + 2∞∑j=1
(j
11
)q j
1 − q j .
POWERS OF THETA FUNCTIONS 5
These are equivalent to instances of a general theorem of Dirichlet; see [Landau1958, Theorem 204]. When k = 1 and p = 23, Theorem 1.2 gives
∞∑m=−∞
∞∑n=−∞
qm2+mn+6n2
= 1 +23
∞∑j=1
(j
23
)q j
1 − q j +43q
∞∏j=1
(1 − q j )(1 − q23 j ),
∞∑m=−∞
∞∑n=−∞
q2m2+mn+3n2
= 1 +23
∞∑j=1
(j
23
)q j
1 − q j −23q
∞∏j=1
(1 − q j )(1 − q23 j ),
and these were proved by F. van der Blij in [1952]. They may be rearranged togive
∞∑m=−∞
∞∑n=−∞
qm2+mn+6n2
+ 2∞∑
m=−∞
∞∑n=−∞
q2m2+mn+3n2
= 3 + 2∞∑j=1
(j
23
)q j
1 − q j ,
∞∑m=−∞
∞∑n=−∞
qm2+mn+6n2
−
∞∑m=−∞
∞∑n=−∞
q2m2+mn+3n2
= 2q∞∏j=1
(1−q j )(1−q23 j ).
The first of these is equivalent to another instance of Dirichlet’s theorem [Landau1958, Theorem 204], and the second formula was noted by J.-P. Serre in [1977,p. 242].
Example 2. For the case p = 3, results for 1 ≤ k ≤ 4 were given (without proof)by Ramanujan [Andrews and Berndt 2005, pp. 402–403], and results for 3 ≤ k ≤ 6were given by H. Petersson in [1982, p. 90]. For 2 ≤ k ≤ 6, these results are:( ∞∑
m=−∞
∞∑n=−∞
qm2+mn+n2
)2
= 1 + 12∞∑j=1
jq j
1 − q j − 36∞∑j=1
jq3 j
1 − q3 j ,( ∞∑m=−∞
∞∑n=−∞
qm2+mn+n2
)3
= 1 − 9∞∑j=1
(j3
)j2q j
1 − q j + 27∞∑j=1
j2q j
1 + q j + q2 j ,( ∞∑m=−∞
∞∑n=−∞
qm2+mn+n2
)4
= 1 + 24∞∑j=1
j3q j
1 − q j + 216∞∑j=1
j3q3 j
1 − q3 j ,( ∞∑m=−∞
∞∑n=−∞
qm2+mn+n2
)5
= 1 + 3∞∑j=1
(j3
)j4q j
1 − q j + 27∞∑j=1
j4q j
1 + q j + q2 j ,( ∞∑m=−∞
∞∑n=−∞
qm2+mn+n2
)6
= 1 +25213
∞∑j=1
j5q j
1 − q j −6804
13
∞∑j=1
j5q3 j
1 − q3 j
+21613 q
∞∏j=1
(1 − q j )6(1 − q3 j )6.
6 HENG HUAT CHAN AND SHAUN COOPER
Results for p = 3, 1 ≤ k ≤ 20, were given by G. Lomadze in [1989a; 1989b].Lomadze’s expansions for 6 ≤ j ≤ 20 are different from ours. For example, Lo-madze’s formula for k = 6 has
112
∞∑n=1
(∑x2
1+x1 y1+y21+x2
2+x2 y2+y22=n
9x41 − 9nx2
1 + n2)
qn
in place of
q∞∏j=1
(1 − q j )6(1 − q3 j )6,
and Lomadze’s formulas become more complicated as k increases.
Example 3. For p = 7, the cases k = 2 and 3 of Theorem 1.2 give
(11)( ∞∑
m=−∞
∞∑n=−∞
qm2+mn+2n2
)2
= 1 + 4∞∑j=1
jq j
1 − q j − 28∞∑j=1
jq7 j
1 − q7 j
and ( ∞∑m=−∞
∞∑n=−∞
qm2+mn+2n2
)3
(12)
= 1 −78
∞∑j=1
(j7
)j2q j
1 − q j +498
∞∑j=1
j2(q j+ q2 j
− q3 j+ q4 j
− q5 j− q6 j )
1 − q7 j
+34q
∞∏j=1
(1 − q j )3(1 − q7 j )3.
The identity (11) was given by Ramanujan; see [Andrews and Berndt 2005,p. 405, Entry 18.2.15]. See [Chan and Ong 1999; Cooper and Toh 2008; Liu2003] and [Williams 2006] for other proofs.
The identity (12) is a consequence of the formulas for E∞
3 (q;χ7) and E03(q;χ7)
in [Chan and Cooper 2008]. In [Chan et al. 2008], it was shown that
q∞∏j=1
(1 − q j )3(1 − q7 j )3 =12
∞∑m=−∞
∞∑n=−∞
(m + n
(1 + i√
72
))2qm2
+mn+2n2.
Another result for z37 can be obtained by combining two of Ramanujan’s results,
[Andrews and Berndt 2005, p. 404, Entry 18.2.14] and [Berndt 1991, p. 467, Entry5 (i)]:
POWERS OF THETA FUNCTIONS 7
( ∞∑m=−∞
∞∑n=−∞
qm2+mn+2n2
)3
(13)
=
∞∏j=1
(1 − q j )7
(1 − q7 j )+ 13q
∞∏j=1
(1 − q j )3(1 − q7 j )3 + 49q2∞∏j=1
(1 − q7 j )7
(1 − q j ).
Other proofs of (13) have been given by H. H. Chan and Y. L. Ong in [1999,Lemma 2.2] and Z.-G. Liu in [2003].
The remainder of this paper is organized as follows. We shall give a proofof Theorem 1.2 in Section 2. The proof depends on three transformation formulas(Lemmas 2.1–2.3) for 00(p), as well as a result that says certain bounded functionsmust be constant (Lemma 2.4). A proof of the identity (13) using the same tech-nique is also given. Some applications to convolution sums are given in Section 3.
2. Proofs
Let
0 =
{(a bc d
): a, b, c, d ∈ Z, ad − bc = 1
},
00(p)=
{(a bc d
): a, b, c, d ∈ Z, ad − bc = 1, c ≡ 0 (mod p)
}.
For p = 3, 7, 11 or 23, define
(14) ηp(τ )= (η(τ )η(pτ))24/(p+1) .
The proof of Theorem 1.2 hinges on the following four lemmas.
Lemma 2.1. Let p = 3, 7, 11 or 23 and let(a b
c d
)∈ 00(p). Then, for ηp(τ ) defined
by (14), we have
ηp
(aτ + bcτ + d
)=
(dp
)24/(p+1)
(cτ + d)24/(p+1)ηp(τ )
andηp
(−1τ√
p
)= (−iτ)24/(p+1)ηp
( τ√
p
).
Proof. These follow from the transformation formula for the Dedekind eta-function[Apostol 1990, p. 52, Theorem 3.4]. �
Lemma 2.2. Let p = 3, 7, 11 or 23 and let(a b
c d
)∈ 00(p). Then, for z p(τ ) defined
by (7), we have
z p
(aτ + bcτ + d
)=
(dp
)(cτ + d)z p(τ )
8 HENG HUAT CHAN AND SHAUN COOPER
and
z p
(−1τ√
p
)= −iτ z p
( τ√
p
).
The same transformation formulas hold when z23 is replaced with z′
23.
Proof. The first result follows from [Schoeneberg 1974, p. 217, Theorem 4] bytaking r = 1, A =
(2 11 (p+1)/2
), h = (0, 0), k = 0 and Pk = 1. The corresponding
result for z′
23 follows by taking A =(4 1
1 6
), with the other parameters being the same
as for the case p = 23.The second result is a direct consequence of [Schoeneberg 1974, p. 205, (5)]. �
Lemma 2.3. Let p ≡ 3 (mod 4) be prime and let n be a positive integer. Let(a bc d
)∈ 00(p). Then for Fk(τ ; p) defined by (4)–(6), we have
Fk
(aτ + bcτ + d
; p)
=
(dp
)k
(cτ + d)k Fk(τ ; p)
and
Fk
(−1τ√
p; p)
= (−iτ)k Fk
( τ√
p; p).
Proof. For odd values of k, these follow from [Cooper 2008, Theorem 6.1] or[Kolberg 1968, (1.8)–(1.12)]. For even values of k with k ≥ 4, these follow fromthe well-known transformation formulas for E2k(τ ), for example, see [Serre 1973,pp. 83, 92, 95–96]. For k = 2, the results are most easily proved by appealing tothe transformation formulas for the function
(η(pτ)η(τ )
)24 in [Apostol 1990, pp. 84–85,Theorems 4.7 and 4.8], and then applying logarithmic differentiation. �
Lemma 2.4. Let f (τ ) be analytic and bounded in the upper half plane Im(τ ) > 0,and suppose it satisfies the transformation property
(15) f(aτ + b
cτ + d
)= f (τ ) for all
(a bc d
)∈ 00(p).
Then f is constant.
Proof. This is Theorem 4.4 in [Apostol 1990, p. 79]. �
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2. Let p = 3, 7, 11 or 23, and let k be a positive integer. Let `be the smallest integer that satisfies 24`
p+1 ≥ k. Consider the functions
ϕ(τ) = ϕp,k(τ )=Fk(τ ; p)(z p(τ ))k
( z p(τ )
η(τ )η(pτ)
)24`/(p+1)and
ψ(τ) = ψp(τ )=
( z p(τ )
η(τ )η(pτ)
)24/(p+1).
POWERS OF THETA FUNCTIONS 9
By Lemmas 2.1–2.3, ϕ(τ) and ψ(τ) satisfy the transformation property (15). Fur-thermore, ϕ and ψ are both analytic in the upper half plane 0 < Im(τ ) < ∞, asη(τ) does not vanish in this region. Let us analyze the behavior at τ = i∞. Fromthe q-expansions, we find that
ϕ(τ)=(1 + O(q))(1 + O(q))k
( 1 + O(q)q + O(q2)
)`= q−`
+ O(q−`+1) as τ → i∞.
Therefore ϕ(τ) has a pole of order ` at i∞. Similarly, we find that ψ(τ) has apole of order 1 at i∞. It follows that there exist constants b1, . . . , b`, such that thefunction
(16) λ(τ) := ϕ(τ)−∑j=1
b j (ψ(τ))j
has no pole at i∞. That is to say,
λ(τ)= b0 + O(q) as τ → i∞
for some constant b0. Let us consider the behavior of λ(τ) at τ = 0. By the secondresult in each of Lemmas 2.1–2.3, we find that
ϕ(
−1τ√
p
)= ϕ
( τ√
p
)and ψ
(−1τ√
p
)= ψ
( τ√
p
).
Therefore
λ(τ)= λ(−1pτ
)−→ b0 as τ → 0.
It follows from the description of the fundamental region for 00(p) given in [Apos-tol 1990, p. 76, Theorem 4.2] that λ(τ) is bounded in the upper half plane. Henceby Lemma 2.4, λ(τ) is constant, that is, λ(τ)≡ b0. Therefore, from (16) we have
ϕ(τ)=
∑j=0
b j (ψ(τ))j .
Using the fact that ψ(τ)= 1/3p(τ ), this is equivalent to
Fk(τ ; p)= zkp
∑j=0
b j3`− jp = zk
p
∑0≤ j≤(p+1)k/24
c j3jp,
where c j = b`− j . Letting q = 0 on both sides we deduce that c0 = 1.If we replace z23 and323 by z′
23 and3′
23, respectively, at every step in the proof,we establish the result for z′
23 and 3′
23.This completes the proof of Theorem 1.2. �
10 HENG HUAT CHAN AND SHAUN COOPER
Remarks. For p = 3, 7, 11 or 23, the genus of the normalizer of 00(p) in SL2(R)
(denoted by00(p)+) is 0. It turns out that for each p, the field of functions invariantunder 00(p)+ is generated by ψp(τ ), which has a simple pole at τ = i∞. Sinceϕp,k(τ ) has a pole of order ` at τ = i∞ and ϕp,k(τ ) is a function on 00(p)+, itfollows that ϕp,k(τ ) is a polynomial in ψp(τ ) with degree exactly `. This explainsthe existence of relation (16).
The identity (13) may be proved similarly.
Proof of (13). Let
F(τ )=z3
7
η3(τ )η3(7τ)and G(τ )=
η4(τ )
η4(7τ).
Lemmas 2.1 and 2.2 imply F(τ ) satisfies the transformation formula (15). Fur-thermore, [Apostol 1990, p. 87, Theorem 4.9] implies that G(τ ) also satisfies thetransformation formula (15). The q-expansions are
(17) F(τ )=1q
+ O(1) and G(τ )=1q
+ O(1) as τ → i∞.
Hence F(τ ) and G(τ ) both have a pole of order 1 at τ = i∞.By the second parts of Lemmas 2.1 and 2.2, and by the transformation formula
for the Dedekind eta-function [Apostol 1990, p. 52, Theorem 3.4], we have
(18) F(−1τ
)= F(τ ) and G
(−1τ
)=
49G(τ )
.
Therefore at the point τ = 0, F(τ ) has a pole of order 1 and G(τ ) has a zero oforder 1.
Let
H(τ ) := F(τ )− aG(τ )−b
G(τ ),
where a and b are constants that will be chosen so that H(τ ) has no pole at 0 ori∞. In order for there to be no pole at τ = i∞, (17) implies a = 1. In order forthere to be no pole at τ = 0, (17) and (18) imply b = 49. It follows that the functionH(τ ) with these values of a and b is bounded in the upper half plane, and Lemma2.4 implies that it is constant. That is,
z37
η3(τ )η3(7τ)= c +
η4(τ )
η4(7τ)+ 49
η4(7τ)η4(τ )
,
for some constant c. If we multiply by η3(τ )η3(7τ) and compare coefficients of qon both sides, we find that c = 13. This completes the proof of (13). �
POWERS OF THETA FUNCTIONS 11
3. Application to convolution sums
Let σ j (n) denote the sum of the j-th powers of the divisors of n, and let σ(n) =
σ1(n). The convolution sum
Wk(n)=
∑1≤m<n/k
σ(m)σ (n − km)
has been evaluated for 1 ≤ k ≤ 14 and k = 16, 18 and 24. See [Alaca et al. 2007]and [Royer 2007] for references. In this section, we show how Theorem 1.2 leadsto an evaluation of Wk(n) for the cases k = 3, 7, 11 and 23. The case k = 23 isnew. Let
P(q)= E2(τ ) = 1 − 24∞∑j=1
jq j
1 − q j ,
Q(q)= E4(τ ) = 1 + 240∞∑j=1
j3q j
1 − q j ,
S(q)= −q24
ddq
P(q) =
∞∑j=1
j2q j
(1 − q j )2.
Theorem 3.1. For p = 3, 7, 11 and 23 we have
P(q)P(q p) =1
p2 + 1(Q(q)+p2 Q(q p))−
144p(S(q)+p2S(q p))−576 z4
pu p(3p),
whereu3(33)= 0,
u7(37)=17037,
u11(311)=1
671(15311 − 173211),
u23(323)=1
2438(77323 − 2223223 + 20133
23 − 303423).
Proof. By Theorem 1.2 with k = 2 and 4, we have
pP(q p)− P(q)p − 1
= z2p
(1 −
∑1≤ j≤(p+1)/12
cp, j3jp
),(19)
p2 Q(q p)+ Q(q)p2 + 1
= z4p
(1 −
∑1≤ j≤(p+1)/6
dp, j3jp
),(20)
for some constants cp, j and dp, j . If we square (19) and subtract the result from(20), we obtain
p2 Q(q p)+ Q(q)p2 + 1
−(pP(q p)− P(q))2
(p − 1)2= z4
p
∑1≤ j≤(p+1)/6
d ′
p, j3jp,
12 HENG HUAT CHAN AND SHAUN COOPER
for some constants d ′
p, j . This may be rewritten as
P(q)P(q p)=1
2p(p2 P2(q p)+ P2(q))−
(p − 1)2
2p(p2 + 1)(p2 Q(q p)+ Q(q))
+z4p
∑1≤ j≤(p+1)/6
d ′′
p, j3jp,
for some constants d ′′
p, j . Now use the result (see [Chan 2007; Glaisher 1885] or[Ramanujan 2000, p. 142, Eq. (30)])
P2(q)= Q(q)− 288S(q)
to get
P(q)P(q p)=1
p2 + 1(Q(q)+ p2 Q(q p))−
144p(S(q)+ p2S(q p))
+z4p
∑1≤ j≤(p+1)/6
d ′′
p, j3jp.
The values of the coefficients d ′′
p, j may be determined by expanding in powers ofq and equating coefficients of q j for 1 ≤ j ≤ (p + 1)/6. In this way we obtainthe polynomials u p(3p) given in the statement of the theorem. This completes theproof. �
Theorem 3.2. For p = 3, 7, 11 and 23 we have
Wp(n)=5
12(p2 + 1)
(σ3(n)+ p2σ3
( np
))+
( 124
−n
4p
)σ(n)+
( 124
−n4
)σ( n
p
)− cp(n).
Here cp(n) is defined by
∞∑n=1
cp(n)qn= z4
pu p(3p),
and u p(3p) is as in Theorem 3.1.
Proof. Equate coefficients of qn on both sides of the identity in Theorem 3.1. �
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14 HENG HUAT CHAN AND SHAUN COOPER
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Received November 6, 2007. Revised December 7, 2007.
HENG HUAT CHAN
DEPARTMENT OF MATHEMATICS
NATIONAL UNIVERSITY OF SINGAPORE
KENT RIDGE 119260SINGAPORE
SHAUN COOPER
INSTITUTE OF INFORMATION AND MATHEMATICAL SCIENCES
MASSEY UNIVERSITY – ALBANY
PRIVATE BAG 102904, NORTH SHORE MAIL CENTRE
AUCKLAND
NEW ZEALAND
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
OPTIMAL OSCILLATION CRITERIA FOR FIRST ORDERDIFFERENCE EQUATIONS WITH DELAY ARGUMENT
GEORGE E. CHATZARAKIS, ROMAN KOPLATADZE
AND IOANNIS P. STAVROULAKIS
Consider the first order linear difference equation
1u(k) + p(k) u(τ(k)) = 0, k ∈ N,
where 1u(k) = u(k + 1) − u(k), p : N → R+, τ : N → N , τ(k) ≤ k − 2 andlimk→+∞ τ(k) = +∞. Optimal conditions for the oscillation of all propersolutions of this equation are established. The results lead to a sharp oscil-lation condition, when k − τ(k) → +∞ as k → +∞. Examples illustratingthe results are given.
1. Introduction
The first systematic study for the oscillation of all solutions to the first order delaydifferential equation
(1-1) u′(t)+ p(t) u(τ (t))= 0,
where
p ∈ L loc(R+; R+), τ ∈ C(R+; R+), τ (t)≤ t for t ∈ R+ and limt→+∞
τ(t)= +∞,
in the case of constant coefficients and constant delays was made by Myshkis[1972]. For the differential equation (1-1) the problem of oscillation is investigatedby many authors. See, for example, [Elbert and Stavroulakis 1995; Koplatadze andChanturiya 1982; Koplatadze and Kvinikadze 1994; Ladas et al. 1984; Sficas andStavroulakis 2003] and the references cited therein.
Theorem 1.1 [Koplatadze and Chanturiya 1982]. Assume that
(1-2) lim inft→+∞
∫ t
τ(t)p(s) ds >
1e.
Then all solutions of Equation (1-1) oscillate.
MSC2000: primary 39A11; secondary 39A12.Keywords: difference equation, proper solution, positive solution, oscillatory.
15
16 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
It is to be emphasized that condition (1-2) is optimal in the sense that it cannotbe replaced by the condition
(1-3) lim inft→+∞
∫ t
τ(t)p(s) ds ≥
1e.
For example, if τ(t)= t−δ or τ(t)=α t or τ(t)= tα, where δ>0, α∈ (0, 1), exam-ples can be given such that condition (1-3) is satisfied, but (1-1) has a nonoscillatorysolution.
The discrete analogue of the first order delay differential equation (1-1) is thefirst order difference equation
(1-4) 1u(k)+ p(k) u(τ (k))= 0,
where
(1-5)1u(k)= u(k + 1)− u(k), p : N → R+,
τ : N → N , τ (k)≤ k − 1, limk→+∞
τ(k)= +∞.
By a proper solution of (1-4) we mean a function u : Nn0 → R with n0 =
min{τ(k) : k ∈ Nn} and Nn = {n, n + 1, . . . }, which satisfies (1-4) on Nn andsup{|u(i)| : i ≥ k}> 0 for k ∈ Nn0 .
A proper solution u : Nn0 → R of (1-4) is said to be oscillatory (around zero) iffor any positive integer n ∈ Nn0 there exist n1, n2 ∈ Nn such that u(n1) u(n2)≤ 0.Otherwise, the proper solution is said to be nonoscillatory. In other words, a propersolution u is oscillatory if it is neither eventually positive nor eventually negative.
Oscillatory properties of the solutions of (1-4), in the case of a general de-lay argument τ(k), have been recently investigated in [Chatzarakis et al. 2008a;2008b], while the special case when τ(k)= k − n, n ≥ 1, has been studied ratherextensively. See, for example, [Agarwal et al. 2005; Bastinec and Diblik 2005;Chatzarakis and Stavroulakis 2006; Domshlak 1999; Elaydi 1999; Ladas et al.1989] and the references cited therein. In this particular case, (1-4) becomes
(1-6) 1u(k)+ p(k) u(k − n)= 0, k ∈ N .
For this equation Ladas, Philos and Sficas established the following theorem.
Theorem 1.2 [Ladas et al. 1989]. Assume that
(1-7) lim infk→+∞
k−1∑i=k−n
p(i) >( n
n + 1
)n+1.
Then all proper solutions of (1-6) oscillate.
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 17
This result is sharp in the sense that the inequality (1-7) cannot be replaced bythe nonstrong one for any n ∈ N . Hence, Theorem 1.2 is the discrete analogue ofTheorem 1.1 when τ(t)= t − δ.
An interesting question then arises whether there exists the discrete analogueof Theorem 1.1 for (1-4) in the case of a general delay argument τ(k) whenlimk→+∞(k − τ(k))= +∞.
In the present paper optimal conditions for the oscillation of all proper solutionsof (1-4) are established and a positive answer to the above question is given.
2. Some auxiliary lemmas
Let k0 ∈ N . Denote by Uk0 the set of all proper solutions of (1-4) satisfying thecondition u(k) > 0 for k ≥ k0.
Remark 2.1. We will suppose that Uk0 = ∅, if (1-4) has no solution satisfying thecondition u(k) > 0 for k ≥ k0.
Lemma 2.2. Assume that k0 ∈ N , Uk0 6= ∅, u ∈ Uk0 , τ(k) ≤ k − 1, τ is a nonde-creasing function and
(2-1) lim infk→+∞
k−1∑i=τ(k)
p(i)= c > 0.
Then
(2-2) lim supk→+∞
u(τ (k))u(k + 1)
≤4c2 .
Proof. By (2-1), for any ε ∈ (0, c), it is clear that
(2-3)k−1∑
i=τ(k)
p(i)≥ c − ε for k ∈ Nk0 .
Since u is a positive proper solution of (1-4), then there exists k1 ∈ Nk0 such that
u(τ (k)) > 0 for k ∈ Nk1 .
Thus, from (1-4) we have
u(k + 1)− u(k)= −p(k)u(τ (k))≤ 0
and so u is an eventually nonincreasing function of positive numbers.Now from inequality (2-3) it is clear that, there exists k∗
≥ k such that
(2-4)k∗
−1∑i=k
p(i) <c − ε
2and
k∗∑i=k
p(i)≥c − ε
2.
18 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
This is because in the case where p(k) < c−ε2 , it is clear that there exists k∗ > k
such that (2-4) is satisfied, while in the case where p(k) ≥c−ε
2 , then k∗= k, and
thereforek∗
−1∑i=k
p(i)=
k−1∑i=k
p(i) (by which we mean) = 0<c − ε
2
andk∗∑
i=k
p(i)=
k∑i=k
p(i)= p(k)≥c − ε
2.
That is, in both cases (2-4) is satisfied.Now, we will show that τ(k∗) ≤ k − 1. Indeed, in the case where p(k) ≥
c−ε2 ,
since k∗= k , it is obvious that τ(k∗) ≤ k − 1. In the case where p(k) < c−ε
2 ,then k∗ > k. Assume, for the sake of contradiction, that τ(k∗) > k − 1. Hence,k ≤ τ(k∗)≤ k∗
− 1 and then
k∗−1∑
i=τ(k∗)
p(i)≤
k∗−1∑
i=k
p(i) <c − ε
2.
This, in view of (2-3), leads to a contradiction. Thus, in both cases, we haveτ(k∗)≤ k − 1.
Therefore, it is clear that
(2-5)k−1∑
i=τ(k∗)
p(i)=
k∗−1∑
i=τ(k∗)
p(i)−k∗
−1∑i=k
p(i)≥ (c − ε)−c − ε
2=
c − ε
2.
Now, summing up (1-4) first from k to k∗ and then from τ(k∗) to k − 1, and usingthat the function u is nonincreasing and the function τ is nondecreasing, we have
u(k)− u(k∗+ 1)=
k∗∑i=k
p(i)u(τ (i))≥
( k∗∑i=k
p(i))
u(τ (k∗))≥c − ε
2u(τ (k∗)),
or
(2-6) u(k)≥c − ε
2u(τ (k∗)),
and then
u(τ (k∗))−u(k)=k−1∑
i=τ(k∗)
p(i)u(τ (i))≥( k−1∑
i=τ(k∗)
p(i))
u(τ (k−1))≥ c−ε
2u(τ (k−1)),
or
(2-7) u(τ (k∗))≥c − ε
2u(τ (k − 1)).
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 19
Combining inequalities (2-6) and (2-7), we obtain
u(τ (k − 1))u(k)
≤4
(c − ε)2
and, for large k, we haveu(τ (k))u(k + 1)
≤4
(c − ε)2.
Hence,
lim supk→+∞
u(τ (k))u(k + 1)
≤4
(c − ε)2,
which, for arbitrarily small values of ε, implies (2-2). �
Lemma 2.3. Assume that k0 ∈ N , Uk0 6= ∅, u ∈ Uk0 , τ(k) ≤ k − 1, τ is a nonde-creasing function and condition (2-1) is satisfied. Then
(2-8) limk→+∞
u(k) exp(λ
k−1∑i=1
p(i))
= +∞ for any λ >4c2 .
Proof. Since all the conditions of Lemma 2.2 are satisfied, for any γ > 4/c2 , thereexists k1 ∈ Nk0 such that
(2-9)u(τ (k))u(k + 1)
≤ γ for k ∈ Nk1 .
Also, for any n ∈ Nk1
n∑k=k1
1u(k)u(k + 1)
=
n∑k=k1
(1 −
u(k)u(k + 1)
)= (n − k1)−
n∑k=k1
exp(
lnu(k)
u(k + 1)
)
≤ (n − k1)−
n∑k=k1
(1 + ln
u(k)u(k + 1)
)= −
n∑k=k1
lnu(k)
u(k + 1)= ln
u(n + 1)u(k1)
,
orn∑
k=k1
1u(k)u(k + 1)
≤ lnu(n + 1)
u(k1).
Moreover, from (1-4), we haven∑
k=k1
1u(k)u(k + 1)
= −
n∑k=k1
p(k)u(τ (k))u(k + 1)
.
Combining (2-9) with the last two relations, we obtain
u(n + 1)≥ u(k1) exp(
−γ
n∑k=k1
p(k)).
20 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
Now, by (2-1), it is obvious that+∞∑
p(i)= +∞. Therefore, for λ > 4/c2, the lastinequality yields
limn→+∞
u(n + 1) exp(λ
n∑k=k1
p(k))
= +∞,
or
limk→+∞
u(k) exp(λ
k−1∑i=k1
p(i))
= +∞,
which implies (2-8), since
k−1∑i=1
p(i)≥
k−1∑i=k1
p(i). �
Next, consider the difference inequality
(2-10) 1u(k)+ q(k) u(σ (k))≤ 0,
whereq : N → R+, σ : N → N and lim
k→+∞
σ(k)= +∞.
In the sequel the following lemma will be used, which has recently been estab-lished in [Chatzarakis et al. 2008a].
Lemma 2.4. Assume that (2-1) is satisfied, and for sufficiently large k
σ(k)≤ τ(k)≤ k − 1, p(k)≤ q(k)
and u : Nk0 → (0,+∞) is a positive proper solution of (2-10). Then, there existsk1 ∈ Nk0 such that Uk1 6= ∅ and u∗ ∈ Uk1 is the solution of (1-4), which satisfies thecondition
0< u∗(k)≤ u(k) for k ∈ Nk1 .
By virtue of Lemma 2.4, we can formulate Lemma 2.3 in the following moregeneral form, where the function τ is not required to be nondecreasing.
Lemma 2.5. Assume that k0 ∈ N , Uk0 6= ∅, u ∈ Uk0 , τ(k) ≤ k − 1 and condition(2-1) is satisfied. Then, for any λ > 4/c2, condition (2-8) holds.
Proof. Since u : Nk0 → (0,+∞) is a solution of (1-4), it is clear that u is a solutionof the inequality
1u(k)+ p(k) u(σ (k))≤ 0 for k ∈ Nk1,
where σ(k) = max{τ(i) : 1 ≤ s ≤ k, s ∈ N } and k1 > k0 is a sufficiently largenumber.
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 21
First we will show that
(2-11) lim infk→+∞
k−1∑i=σ(k)
p(i)= c.
Assume that (2-11) is not satisfied. Then there exists a sequence {ki }+∞
i=1 of naturalnumbers such that σ(ki ) 6= τ(ki ) (i = 1, 2, . . . ) and
(2-12) lim infj→+∞
k j −1∑i=σ(k j )
p(i)= c1 < c.
Also, from the definition of the function σ , and in view of σ(ki ) 6= τ(ki ), for anyki , there exists k ′
i < ki such that σ(k)= σ(ki ) for k ′
i ≤ k ≤ ki , limi→+∞ k ′
i = +∞
and σ(k ′
i )= τ(k ′
i ). Thus
k′
i −1∑j=τ(k′
i )
p( j)=
k′
i −1∑j=σ(k′
i )
p( j)=
k′
i −1∑j=σ(ki )
p( j)≤
ki −1∑j=σ(ki )
p( j) (i = 1, 2, . . . ),
and, by the virtue of (2-12), we have
lim infi→+∞
k′
i −1∑j=τ(k′
i )
p( j)≤ lim infi→+∞
ki −1∑j=σ(ki )
p( j)= c1 < c.
In view of (2-1), the last inequality leads to a contradiction. Therefore (2-11) holds.Now, by Lemma 2.4, we conclude that the equation
1u(k)+ p(k) u(σ (k))= 0
has a solution u∗ which satisfies the condition
(2-13) 0< u∗(k)≤ u(k) for k ∈ Nk1,
where k1 > k0 is a sufficiently large number. Hence, taking into account that thefunction σ is nondecreasing, in view of Lemma 2.3, we have
limk→+∞
u∗(k) exp(λ
k−1∑i=1
p(i))
= +∞,
where λ > 4/c2. Therefore, by (2-13), we get
limk→+∞
u(k) exp(λ
k−1∑i=1
p(i))
= +∞ for any λ >4c2 . �
22 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
Lemma 2.6 (Abel transformation). Let {ai }+∞
i=1 and {bi }+∞
i=1 be sequences of non-negative numbers and
(2-14)+∞∑i=1
ai <+∞.
Thenk∑
i=1
ai bi = A1 b1 − Ak+1 bk+1 −
k∑i=1
Ai+1(bi − bi+1),
where Ai =∑
+∞
j=i a j .
Proof. Since (2-14) is satisfied, we have
k∑i=1
Ai+1(bi − bi+1)=
k∑i=1
Ai+1bi −
k+1∑i=2
Ai bi
= A2b1 − Ak+1bk+1 +
k∑i=2
(Ai+1 − Ai )bi
= A2b1 − Ak+1bk+1 −
k∑i=2
ai bi
= A1b1 − Ak+1bk+1 −
k∑i=1
ai bi ,
ork∑
i=1
ai bi = A1b1 − Ak+1bk+1 −
k∑i=1
Ai+1(bi − bi+1). �
Koplatadze, Kvinikadze and Stavroulakis established the following lemma. Forcompleteness, we present the proof here.
Lemma 2.7 [Koplatadze et al. 2002]. Let ϕ,ψ : N → (0,+∞), ψ be nonincreas-ing and suppose
limk→+∞
ϕ(k)= +∞,(2-15)
lim infk→+∞
ψ(k) ϕ(k)= 0,(2-16)
where ϕ(k)= inf{ϕ(s) : s ≥ k, s ∈ N }. Then there exists an increasing sequence ofnatural numbers {ki }
+∞
i=1 such that
limi→+∞
ki = +∞, ϕ(ki )= ϕ(ki ), ψ(k) ϕ(k)≥ ψ(ki ) ϕ(ki )
(k = 1, 2, . . . , ki ; i = 1, 2, . . .).
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 23
Proof. Define the sets E1 and E2 by
k ∈ E1 ⇐⇒ ϕ(k)= ϕ(k),
k ∈ E2 ⇐⇒ ϕ(s) ψ(s)≥ ϕ(k) ψ(k) for s ∈ {1, . . . , k}.
According to (2-15) and (2-16), it is obvious that
(2-17) sup Ei = +∞ (i = 1, 2).
Show that
(2-18) sup E1 ∩ E2 = +∞.
Let k0 ∈ E2 be such that k0 /∈ E1. By (2-16) there is k1 > k0 such that ϕ(k)= ϕ(k1)
for k = k0, k0 + 1, . . . , k1 and ϕ(k1)= ϕ(k1). Since ψ is nonincreasing, we have
ϕ(k) ψ(k)≥ ϕ(k1) ψ(k1) for k = 1, . . . , k1.
Therefore k1 ∈ E1∩E2. The above argument together with (2-17) imply that (2-18)holds. �
Remark 2.8. The analogue of this lemma for continuous functions ϕ and ψ wasproved first in [Koplatadze 1994].
3. Necessary conditions of the existence of positive solutions
The results of this section play an important role in establishing sufficient condi-tions for all proper solutions of (1-4) to be oscillatory.
Theorem 3.1. Assume that k0 ∈ N , Uk0 6= ∅, (1-5) is satisfied,
(3-1) lim infk→+∞
k−1∑i=τ(k)
p(i)= c > 0,
and
(3-2) lim supk→+∞
k−1∑i=τ(k)
p(i) <+∞.
Then there exists λ ∈ [1, 4/c2] such that
(3-3)
lim supε→0+
(lim infk→+∞
exp((λ+ ε)
k−1∑i=1
p(i)) +∞∑
i=k
p(i) exp(
−(λ+ ε)
τ(i)−1∑l=1
p(l)))
≤ 1.
24 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
Proof. Since Uk0 6= ∅, Equation (1-4) has a positive solution u : Nk0 → (0,+∞).First we show that
(3-4) lim supk→+∞
u(k) exp( k−1∑
i=1
p(i))<+∞.
Indeed, if k1 ∈ Nk0 , we have
k∑i=k1
1u(i)u(i)
=
k∑i=k1
u(i + 1)u(i)
− (k − k1) =
k∑i=k1
exp(
lnu(i + 1)
u(i)
)− (k − k1)
≥
k∑i=k1
(1 + ln
u(i + 1)u(i)
)− (k − k1) = ln
u(k + 1)u(k1)
,
ork∑
i=k1
1u(i)u(i)
≥ lnu(k + 1)
u(k1).
By (1-4), and taking into account that the function u is nonincreasing, we have
k∑i=k1
1u(i)u(i)
= −
k∑i=k1
p(i)u(τ (i))
u(i)≤ −
k∑i=k1
p(i).
Combining the last two inequalities, we obtain
u(k + 1) exp( k∑
i=k1
p(i))
≤ u(k1),
that is, (3-4) is fulfilled. On the other hand, since all the conditions of Lemma 2.5are satisfied, we conclude that condition (2-8) holds for any λ > 4/c2. Denote by3 the set of all λ for which
(3-5) limk→+∞
u(τ (k)) exp(λ
τ(k)−1∑i=1
p(i))
= +∞
and λ0 = inf3. In view of (1-5), (2-8) and (3-4), it is obvious that λ0 ∈ [1, 4/c2].
Thus, it suffices to show, that for λ = λ0 the inequality (3-3) holds. First, we willshow that for any ε > 0
(3-6) limk→+∞
u(τ (k)) exp((λ0 + ε)
τ(k)−1∑i=1
p(i))
= +∞.
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 25
Indeed, if λ0 ∈ 3, it is obvious from (3-5) that condition (3-6) is fulfilled. Ifλ0 6∈ 3, according to the definition of λ0, there exists λk > λ0 such that λk → λ0
when k → +∞ and λk ∈ 3 , k = 1, 2, . . . . Thus, condition (3-5) holds for anyλ= λk . However, for any ε > 0, there exists λk = λk(ε) such that λ0 <λk ≤ λ0 +ε.This insures the validity of (3-5) and (3-6) for any ε > 0.
Similarly, we show that for any ε > 0,
(3-7) lim infk→+∞
u(τ (k)) exp((λ0 − ε)
τ(k)−1∑i=1
p(i))
= 0 .
Hence, by virtue of (1-5), (3-6) and (3-7), it is clear that for any ε>0, the functions
(3-8) ϕ(k)= u(τ (k)) exp((λ0 + ε)
τ(k)−1∑i=1
p(i))
and
ψ(k)= exp(
−2εk−1∑i=1
p(i))
satisfy the conditions of Lemma 2.7 for sufficiently large k. Hence, there exists anincreasing sequence {ki }
+∞
i=1 of natural numbers satisfying limi→+∞ ki = +∞,
(3-9) ψ(ki ) ϕ(ki )≤ ψ(k) ϕ(k) for k∗≤ k ≤ ki ,
where k∗ is a sufficiently large number, and
(3-10) ϕ(ki )= ϕ(ki ) (i = 1, 2, . . . ),
Now, given that
u(τ (i)) exp((λ0+ε)
τ(i)−1∑l=1
p(l))
≥ inf{
u(τ (s)) exp(λ0+ε)
τ(s)−1∑l=1
p(l) : s ≥ i, s ∈ N}
= ϕ(i),
Equation (1-4) implies
u(τ (k j ))≥
+∞∑i=τ(k j )
p(i) u(τ (i))≥
+∞∑i=τ(k)
p(i) ϕ(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
26 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
that is,
u(τ (k j ))≥
k j −1∑i=τ(k j )
p(i) ϕ(i) exp(
−2εi−1∑l=1
p(l))
exp(
2εi−1∑l=1
p(l))
× exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
+
+∞∑i=k j
p(i) ϕ(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l)),
for j = 1, 2, . . . . Thus, by (3-9), and using the fact that the function ϕ is non-decreasing, the last inequality yields
(3-11) u(τ (k j ))≥ ϕ(k j ) exp(
−2εk j −1∑l=1
p(l))
×
k j −1∑i=τ(k j )
p(i) exp(
2εi−1∑l=1
p(l))
exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
+ ϕ(k j )
+∞∑i=k j
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
( j = 1, 2, . . . ).
Also, in view of Lemma 2.6, we have
I (k j , ε)=
k j −1∑i=τ(k j )
p(i) exp(
2εi−1∑l=1
p(l))
exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
(3-12)
= exp(
2ετ(k j )−1∑
i=1
p(i)) +∞∑
i=τ(k j )
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
− exp(
2εk j −1∑i=1
p(i)) +∞∑
i=k j
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
+
k j −1∑i=k j
(exp
(2ε
i∑l=1
p(l))
− exp(
2εi−1∑l=1
p(l)))
×
+∞∑i=1
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
( j = 1, 2, . . . ).
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 27
Given that
exp(
2εi∑
l=1
p(l))
− exp(
2εi−1∑l=1
p(l))
≥ 0,
inequality (3-12) becomes
I (k j , ε) ≥ exp(
2ετ(k j )−1∑
i=1
p(i)) +∞∑
i=τ(k j )
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
− exp(
2εk j −1∑i=1
p(i)) +∞∑
i=k j
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l)).
Therefore, by (3-11), we take
u(τ (k j ))≥ ϕ(k j ) exp(
−2εk j −1∑l=1
p(l))
exp(
2ετ(k j )−1∑
l=1
p(l))
×
+∞∑i=τ(k j )
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l)).
Thus, (3-8) and (3-10) imply
exp((λ0 + ε)
τ(k j )−1∑i=1
p(i)) +∞∑
i=τ(k j )
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))≤ exp
(2ε
k j −1∑i=τ(k j )
p(i)).
From the last inequality, and taking into account that (3-2) is satisfied, we have
(3-13) lim supj→+∞
exp((λ0 +ε)
τ(k j )−1∑i=1
p(i)) +∞∑
i=τ(k j )
p(i) exp(
−(λ0 +ε)
τ(i)−1∑l=1
p(l))
≤ exp(2εM),
where
M = lim supk→+∞
k−1∑i=τ(k)
p(i).
Hence, for any ε > 0, (3-13) gives
lim infk→+∞
exp((λ0 + ε)
k−1∑i=1
p(i)) +∞∑
i=k
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
≤ exp(2εM),
28 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
which implies
lim supε→0+
(lim infk→+∞
exp((λ0 + ε)
k−1∑i=1
p(i)) +∞∑
i=k
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l)))
≤ 1.
�
Remark 3.2. Condition (3-2) is not a limitation since, as proved in [Chatzarakiset al. 2008a], if τ is a nondecreasing function and
lim supk→+∞
k∑i=τ(k)
p(i) > 1,
then Uk0 = ∅, for any k0 ∈ N .
Remark 3.3. In (3-1), without loss of generality, we may assume that c ≤ 1. Oth-erwise, for any k0 ∈ N , we have Uk0 = ∅ [Chatzarakis et al. 2008a].
Theorem 3.4. Assume that all the conditions of Theorem 3.1 are satisfied. Then
(3-14) lim infk→+∞
k−1∑i=τ(k)
p(i)≤1e.
Proof. Since all the conditions of Theorem 3.1 are satisfied, there exists λ = λ0 ∈
[1, 4/c2] such that the inequality (3-3) holds.
Assume that the condition (3-14) does not hold. Then, there exists k1 ∈ N andε0 > 0 such that
k−1∑i=τ(k)
p(i)≥1 + ε0
efor k ∈ Nk1 .
Therefore, for any ε > 0,
(3-15) I (k, ε)= exp((λ0 + ε)
k−1∑i=1
p(i)) +∞∑
i=k
p(i) exp(
−(λ0 + ε)
τ(i)−1∑l=1
p(l))
≥ exp((λ0 + ε)(1 + ε0)
e
)exp
((λ0 + ε)
k−1∑i=1
p(i))
×
+∞∑i=k
p(i) exp(
−(λ0 + ε)
i−1∑l=1
p(l))
for k ∈ Nk1 .
Defining∑i−1
l=1 p(l)= ai−1, we will show that
lim infk→+∞
exp((λ0 + ε)ak−1)
+∞∑i=k
p(i) exp(−(λ0 + ε)ai−1)≥1
λ0 + ε.
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 29
Indeed, since
lim infk→+∞
k−1∑i=τ(k)
p(i)= c > 0,
it is obvious that∑
+∞
i=1 p(i)= +∞, that is, limi→+∞ ai = +∞. Therefore
exp((λ0 + ε)ak−1)
+∞∑i=k
p(i) exp(−(λ0 + ε)ai−1)
= exp((λ0 + ε)ak−1)
+∞∑i=k
(ai − ai−1) exp(−(λ0 + ε)ai−1)
= exp((λ0 + ε)ak−1)
+∞∑i=k
exp(−(λ0 + ε)ai−1)
∫ ai
ai−1
ds
≥ exp((λ0 + ε)ai−1)
+∞∑i=k
∫ ai
ai−1
exp(−(λ0 + ε)s)ds
= exp((λ0 + ε)ai−1)
∫+∞
ai−1
exp(−(λ0 + ε)s)ds =1
λ0 + ε.
Hence, by (3-15), we obtain
lim supε→0+
(lim infk→+∞
I (k, ε))
≥1λ0
· exp(λ0(1 + ε0)
e
)≥ 1 + ε0 .
This contradicts (3-3) for λ= λ0. �
4. Sufficient conditions of the proper solutions to be oscillatory
Theorem 4.1. Assume that conditions (1-5), (3-1), (3-2) are satisfied and that, forany λ ∈ [1, 4/c2
],
(4-1)
lim supε→0+
(lim infk→+∞
(exp
((λ+ε)
k−1∑i=1
p(i)) +∞∑
i=k
p(i) exp(
−(λ+ε)
τ(i)−1∑l=1
p(l)i)))
>1.
Then all proper solutions of Equation (1-4) oscillate.
Proof. Assume that u : Nk0 → (0,+∞) is a positive proper solution of (1-4). ThenUk0 6= ∅. Thus, in view of Theorem 3.1, there exists λ0 ∈ [1, 4/c2
] such that thecondition (3-3) is satisfied for λ= λ0. But this contradicts (4-1). �
Using Theorem 3.4, we can similarly prove:
30 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
Theorem 4.2. Assume that conditions (1-5) and (3-2) are satisfied and
(4-2) lim infk→+∞
k−1∑i=τ(k)
p(i) >1e.
Then all proper solutions of Equation (1-4) oscillate.
Remark 4.3. It is to be pointed out that Theorem 4.2 is the discrete analogue ofTheorem 1.1 for the first order difference equation (1-4) in the case of a generaldelay argument τ(k).
Remark 4.4. The condition (4-2) is optimal for (1-4) under the assumption that
limk→+∞
(k − τ(k))= +∞,
since in this case the set of natural numbers increases infinitely in the interval[τ(k), k − 1] for k → +∞.
Now, we are going to present two examples to show that the condition (4-2) isoptimal, in the sense that it cannot be replaced by the nonstrong inequality.
Example 4.5. Consider (1-4), where
(4-3)τ(k)= [αk], p(k)= (k−λ
− (k + 1)−λ)[αk]λ,
α ∈ (0, 1), λ= − ln−1 α,
with [αk] the integer part of αk.It is obvious that
k1+λ(k−λ− (k + 1)−λ)→ λ for k → +∞ .
Therefore
(4-4) k(k−λ− (k + 1)−λ)[αk]
λ→
λ
efor k → +∞ .
Hence, in view of (4-3) and (4-4), we have
lim infk→+∞
k−1∑i=τ(k)
p(i)=λ
elim infk→+∞
k−1∑i=[αk]
eλ
i(i−λ− (i + 1)−λ)[αi]λ
1i
=λ
elim infk→+∞
k−1∑i=[αk]
1i
=λ
eln
1α
=1e
or
lim infk→+∞
k−1∑i=τ(k)
p(i)=1e.
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 31
Observe that all the conditions of Theorem 4.2 are satisfied except the condition(4-2). In this case, it is not guaranteed that all solutions of (1-4) oscillate. Indeed,it is easy to see that the function u = k−λ is a positive solution of (1-4).
Example 4.6. Consider (1-4), where
(4-5)τ(k)= [kα], p(k)= (ln−λ k − ln−λ(k + 1)) lnλ[kα],
α ∈ (0, 1), λ= − ln−1 α,
with [kα] the integer part of kα.It is obvious that
k ln1+λ k(ln−λ k − ln−λ(k + 1))→ λ for k → +∞ .
Therefore
(4-6) k ln k lnλ[kα](ln−λ k − ln−λ(k + 1))→λ
efor k → +∞ .
On the other hand,
k−1∑i=[kα]
1i ln i
≥
k−1∑i=[kα]
∫ i+1
i
dss ln s
=
∫ k
[kα]
dss ln s
= lnln k
ln[kα],
which tends to ln(1/α) as k → +∞, and
k−1∑i=[kα]
1i ln i
≤
k−1∑i=[kα]
∫ i
i−1
dss ln s
=
∫ k−1
[kα]−1
dss ln s
= lnln(k − 1)ln[kα] − 1
,
which also tends to ln(1/α) as k → +∞. Together these two bounds imply
limk→+∞
k−1∑i=[kα]
1i ln i
= ln1α.
Hence, in view of (4-5) and (4-6), we obtain
lim infk→+∞
k−1∑i=[kα]
p(i)= lim infk→+∞
k−1∑i=[kα]
lnλ[iα](ln−λ i − ln−λ(i + 1))
=λ
elim infk→+∞
k−1∑i=[kα]
eλ
i ln i lnλ[iα](ln−λ i − ln−λ(i + 1))1
i ln i
=λ
elim infk→+∞
k−1∑i=[kα]
1i ln i
=λ
eln
1α
=1e.
32 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS
We again observe that all the conditions of Theorem 4.2 are satisfied except (4-2).In this case, it is not guaranteed that all solutions of (1-4) oscillate. Indeed, it iseasy to see that the function u = ln−λ k is a positive solution of (1-4).
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[Myshkis 1972] A. D. Myshkis, Lineinye differencialnye uravneni� s zapazdyva�-wim argumentom, 2nd ed., Nauka, Moscow, 1972. MR 50 #5135 Zbl 0261.34040
OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 33
[Sficas and Stavroulakis 2003] Y. G. Sficas and I. P. Stavroulakis, “Oscillation criteria for first-orderdelay equations”, Bull. London Math. Soc. 35:2 (2003), 239–246. MR 2003m:34160 Zbl 1035.34075
Received May 30, 2007. Revised July 10, 2007.
GEORGE E. CHATZARAKIS
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF IOANNINA
451 10 IOANNINA
GREECE
ROMAN KOPLATADZE
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF TBILISI
UNIVERSITY STREET 2TBILISI 0143GEORGIA
IOANNIS P. STAVROULAKIS
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF IOANNINA
451 10 IOANNINA
GREECE
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
GENERALIZED HANDLEBODY SETS AND NON-HAKEN3-MANIFOLDS
JESSE EDWARD JOHNSON AND TERK PATEL
In the curve complex for a surface, a handlebody set is the set of loops thatbound properly embedded disks in a given handlebody bounded by the sur-face. A boundary set is the set of nonseparating loops in the curve complexthat bound two-sided, properly embedded surfaces. For a Heegaard split-ting, the distance between the boundary sets of the handlebodies is zeroif and only if the ambient manifold contains a nonseparating, two sidedincompressible surface. We show that every vertex in the curve complex iswithin two edges of a point in the boundary set.
1. Introduction
The curve complex C(6) for a compact, connected, closed, orientable surface 6is the simplicial complex whose vertices are loops (isotopy classes of essential,simple closed curves) in 6 and whose simplices correspond to sets of pairwisedisjoint loops in 6. Given a handlebody H and a homeomorphism φ : 6 → ∂H ,we can define the following subsets of C(6).
The handlebody set H is the set of loops that bound properly embedded (essen-tial) disks in H . The genus g boundary set Hg is the set of nonseparating loopssuch that each bounds a properly embedded, two-sided, incompressible, genus-gsurface in H . Note that H0 is a proper subset of H, specifically the set of all thenonseparating loops in H. Define the boundary set to be the union
H∞=
⋃g≥0
Hg.
We will say that a set A of vertices in C(6) is k-dense if every vertex in C(6) iswithin k edges of a point in A.
Theorem 1. If 6 has genus 3 or greater, then H∞ is 2-dense in C(6).
MSC2000: 57M50.Keywords: curve complex, non-Haken 3-manifold.Research supported by NSF MSPRF grant 0602368.
35
36 JESSE EDWARD JOHNSON AND TERK PATEL
The proof presented here does not work for genus two surfaces. However,Schleimer has shown in [2005] that the orbit of a vertex of C(6) under the actionof the Torelli group is 5-dense. This implies that for a genus two handlebody, H∞
is n-dense for some n ≤ 5.In contrast to H∞, a fixed genus boundary set Hg has a geometric structure much
closer to H, which is not k-dense for any k. This is demonstrated by the followingtwo Lemmas, the first of which is a corollary of Conclusion III.15 in [Jaco 1980]and the second of which follows from a Theorem of Scharlemann [2006].
Lemma 2. If 6 has genus three or greater and v ∈ H then
d(v,Hg)= 1
for every g > 0. If 6 has genus two then
d(v,Hg) > 1
for every g.
Lemma 3. For g ≥ 1, the set Hg is disjoint from H and contained in a 2g neigh-borhood of H.
For this paper, every 3-manifold will be compact, connected, closed and ori-entable. A Heegaard splitting for such a 3-manifold M is a triple (6, H1, H2)
where 6⊂ M is a compact, connected, closed, orientable surface and H1, H2 ⊂ Mare handlebodies such that
∂H1 =6 = ∂H2 and M = H1 ∪ H2.
The inclusion maps from ∂H1 and ∂H2 onto 6 determine handlebody sets H1 andH2, respectively. The distance of the Heegaard splitting, as defined by Hempel in[2001] is the distance d(6)= d(H1,H2) between the two handlebody sets.
The inclusion maps also determine boundary sets Hg1 , Hh
2 , H∞
1 , H∞
2 , allowingus to generalize this distance to the (g, h)-distance
dg,h(6)= d(Hg1,Hh
2)
and the boundary distance
d∞(6)= d(H∞
1 ,H∞
2 ).
The set H∞ is precisely the set of vertices representing simple closed curveswhose homology class is nontrivial in6, but trivial in H . For a Heegaard splitting,it encodes homology information about the ambient manifold. In particular, theboundary distance determines precisely when a manifold has infinite homology(and therefore a nonseparating, incompressible surface).
GENERALIZED HANDLEBODY SETS AND NON-HAKEN 3-MANIFOLDS 37
Lemma 4. The following are equivalent:(1) the first homology group of M is infinite;(2) M contains a nonseparating, two sided, closed incompressible surface;(3) d∞(6)= 0 and(4) d 0,∞(6)= 0.
The proof is given in Section 2. The equivalence of (1) and (2) is well known,but we give a very simple, geometric proof via the boundary set. Theorem 1 isproved in Section 3.
For any Heegaard splitting (6, H1, H2) of a non-Haken 3-manifold, Lemma 4implies that the boundary set in C(6) determined by H2 must be completely dis-joint from the boundary set for H1. Hempel showed that there are handlebody setsthat are arbitrarily far apart in the curve complex. The same is not true for boundarysets. In particular, Theorem 1 implies that for any Heegaard splitting (6, H1, H2)
of genus 3 or greater, d∞(6) is equal to either 0, 1 or 2. For non-Haken manifolds,we have the following.
Corollary 5. For any Heegaard splitting (6, H1, H2) of a non-Haken 3-manifoldM , d∞(6) is equal to 1 or 2.
2. Nonseparating surfaces
The following Lemma will not be used until Section 3, but the method of proofgives a good introduction to the proof of Lemma 7. Recall that an element α ofa Z module G is called primitive if there is no β ∈ G such that α = kβ for somek 6= ±1.
Lemma 6. Let `1, . . . , `k be pairwise disjoint, essential loops in the boundary ofa genus-g handlebody H with g> k. Then there is a properly embedded, nonsepa-rating surface F ⊂ H such that ∂F is disjoint from each ` j and the homology classdefined by ∂F in H(6) is primitive.
Proof. Let D1, . . . , Dg be a system of disks for H , that is, a collection of properlyembedded, essential disks whose complement in H is a single ball. Orient theboundaries of the disks and the loops `1, . . . , `k , then form the matrix A = (ai j )
such that ai j is the algebraic intersection number of Di and ` j .If we replace one of the disks in the system by a disk slide, the matrix for the
new system of disks can be constructed from A by adding or subtracting one rowfrom the other. Thus we can perform elementary row operations on A by choosingnew systems of disks for H . In particular, we can make A upper triangular.
Because A has more rows than columns, if A is upper triangular then the bottomrow consists of all zeros. In other words, the disk Dg has algebraic intersection 0with each loop ` j .
38 JESSE EDWARD JOHNSON AND TERK PATEL
If Dg intersects the loop `1, there must be a pair of adjacent intersections in`1 with opposite orientations. By attaching a band from ∂Dg to itself, along thearc of the loop `1, we can form a new surface F1 whose boundary (consisting oftwo loops) has algebraic intersection zero with each loop ` j , but whose geometricintersection number with the collection of curves `1, . . . , `k is strictly lower thanthat of Dg. The surface F1 is two sided because the intersections have oppositeorientations and nonseparating because Dg is nonseparating.
If ∂F1 intersects `1, we can form a new surface F2 by attaching a band, and soon. Continuing in this manner for each ` j , we form a surface F which is properlyembedded, two-sided, nonseparating and such that ∂F is disjoint from each ` j .
Attaching a band to the boundary of Fi does not change the homology class ofthe boundary, so the homology class of ∂F is equal to the class of ∂Dg. Because∂Dg is represented by a connected loop, its homology class is primitive, as is thehomology class of ∂F . �
We will now use the idea of attaching bands to eliminate intersections to provethe implication (1)⇒ (4) of Lemma 4.
Lemma 7. If the first homology of M is infinite then d 0,∞(6)= 0.
Proof. Let D1, . . . , Dg be a system of disks for H1 and D′
1, . . . , D′g be a system of
disk for H2. Orient the boundaries of both systems of disks. Let A be the matrixof algebraic intersection numbers of the boundaries. Because the first homology isinfinite, the determinant of A must equal zero.
As in the last proof, we can perform row operations on A by taking disk slidesof the disks D1, . . . , Dg. Because the determinant of A is zero, some sequence ofdisk slides will leave A with all zeros in the bottom row. Thus after a sequence ofdisk slides, we can assume Dg has algebraic intersection 0 with each D′
j .By attaching bands to the boundary of Dg as in the proof of Lemma 6, we can
form a properly embedded, two sided, nonseparating surface F whose boundary isdisjoint from D′
1, . . . , D′g. Thus each boundary component of F bounds a disk in
H2. The union of F and these disks is a properly embedded, two sided, nonsepa-rating closed surface in M .
Recall that F was constructed from Dg by attaching bands to its boundary.The last band defines a boundary compression for F corresponding to an isotopypushing this last band into H2. After this isotopy, the second to last band definesa second isotopy, and so on. The final result is a surface isotopic to F whichintersects H1 in a disk isotopic to Dg.
The intersection of this surface with H2 is orientable, two-sided and nonsepa-rating because F has these properties. Thus ∂Dg is in both H0
1 and H∞
2 so
d 0,∞(6)= d(H01,H∞
2 )= 0. �
GENERALIZED HANDLEBODY SETS AND NON-HAKEN 3-MANIFOLDS 39
Proof of Lemma 4. Lemma 7 implies that for any Heegaard splitting (6, H1, H2),
d(H01,H∞
2 )= 0
so (1)⇒ (4). Because H01 is contained in H∞
1 , (4)⇒ (3) is immediate.Let (6, H1, H2) be a Heegaard splitting for M . If d∞(6) = 0 then there is a
simple closed curve ` ⊂ 6 such that ` bounds two-sided, nonseparating properlyembedded surfaces F ⊂ H1 and F ′
⊂ H2. The union F ∪F ′ is a two-sided, nonsep-arating closed surface embedded in M . Compressing F∪F ′ to either side producesat least one new two-sided, nonseparating surface. By compressing repeatedly, weeventually find a closed, nonseparating, two-sided incompressible surface in M .Thus (3)⇒ (2).
The final step, (2)⇒ (1), is a classical result. If M contains a two-sided, non-separating, closed surface S ⊂ M , let p be a point in S. There is a path
α : [0, 1] → M
from p to itself that does not cross S. The homology class of α has infinite orderso the first homology of M is infinite. �
3. Density
Proof of Theorem 1. We will prove the following: Let ` be a loop in ∂H andassume the genus of H is at least 3. Then there is an essential loop `′ disjoint from` and a properly embedded, two-sided, nonseparating surface F such that ∂F is asingle, nonseparating loop disjoint from `′.
By Lemma 6, there is a properly embedded surface F ′′⊂ H such that ∂F ′′ is
disjoint from ` and defines a primitive element of the homology. Of all the properlyembedded surfaces with boundary disjoint from ` and homologous to ∂F ′′, let F ′
be one with minimal number of boundary components. Each component of ∂F ′
has an orientation induced by F ′ and thus defines an element of the first homologyof 6.
For each component C of 6 \ (`∪∂F ′), an orientation for a loop in ∂C inducesan orientation of C . Assume two components of ∂C come from loops of ∂F ′ andinduce the same orientation of C . Because the induced orientations agree, addinga band between them produces a new orientable surface with fewer boundary com-ponents, but homologous boundary. Thus the minimality assumption implies thateach component C of 6 \ (`∪ ∂F ′) has at most one boundary loop coming from∂F ′ inducing each possible orientation. Thus it has at most two boundary loopscoming from ∂F ′ and these induce opposite orientations on C .
Assume for contradiction each component of 6 \ (` ∪ ∂F ′) is planar. Eachcomponent that is disjoint from ` has exactly two boundary components. A planar
40 JESSE EDWARD JOHNSON AND TERK PATEL
surface with two boundary loops is an annulus so each component disjoint from `
must be an annulus. There are either two components of 6 \ (` ∪ ∂F ′) with oneboundary loop each on `, or one component with two boundary loops on `. In thefirst case, the two components are pairs of pants or annuli, while in the second,the component is a four punctured sphere or a pair of pants. The union of suchcomponents and a collection of annuli is a genus-one or genus-two surface. Thiscontradicts the assumption that 6 has genus at least three, so we conclude thatsome component must be nonplanar.
Let C be a nonplanar component. There is a simple closed curve `′ ⊂ C suchthat `′ separates a once-punctured torus from C . In6, the loop `′ separates a once-punctured torus that contains no components of ∂F ′. Let F be a surface whoseboundary is homologous to F ′, disjoint from the once-punctured torus boundedby `′ and such that the number of boundary components of F ′ is minimal over allsuch surfaces.
Once again, each component of 6 \ (`′ ∪ ∂F) has at most two boundary com-ponents on loops in ∂F , with opposite induced orientations. Because `′ bounds asurface disjoint from ∂F , each component of 6 \ ∂F must also have at most twoboundary loops on ∂F .
If a component C of 6 \ ∂F has a single boundary component, this loop ishomology trivial in 6. Attaching a boundary parallel surface to F removes thisloop so minimality of ∂F implies that each component has two boundary loops.
If C has two boundary loops (with opposite induced orientations) then theseloops determine the same element of the homology of 6. Because 6 is connected,this implies that any two loops of ∂F (with their induced orientations) determinethe same element of the homology. Thus the element of the homology determinedby ∂F is of the form kβ where k is the number of boundary components of F .
By Lemma 6, ∂F determines a primitive element of the homology of 6, so kmust be 1. In other words, the boundary of F is connected and ∂F determines anelement of H∞. By construction, ∂F is disjoint from a loop `′ that is disjoint from`. Thus the vertex v ∈ C(6) determined by ` is distance at most 2 from H∞. �
References
[Hempel 2001] J. Hempel, “3-manifolds as viewed from the curve complex”, Topology 40:3 (2001),631–657. MR 2002f:57044 Zbl 0985.57014
[Jaco 1980] W. Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series inMathematics 43, American Mathematical Society, Providence, R.I., 1980. MR 81k:57009 Zbl0433.57001
[Scharlemann 2006] M. Scharlemann, “Proximity in the curve complex: boundary reduction andbicompressible surfaces”, Pacific J. Math. 228:2 (2006), 325–348. MR 2008c:57035 Zbl 05190398
[Schleimer 2005] S. Schleimer, “Notes on the complex of curves”, lecture notes, 2005, Available athttp://www.math.rutgers.edu/~saulsch/math.html.
GENERALIZED HANDLEBODY SETS AND NON-HAKEN 3-MANIFOLDS 41
Received July 11, 2007. Revised September 27, 2007.
JESSE EDWARD JOHNSON
MATHEMATICS DEPARTMENT
YALE UNIVERSITY
PO BOX 208283NEW HAVEN, CT 06520-8283UNITED STATES
[email protected]://math.yale.edu/~jj327/
TERK PATEL
18 DUMAS STREET
PONDICHERRY - 605001INDIA
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
LEFT-SYMMETRIC SUPERALGEBRAIC STRUCTURES ONTHE SUPER-VIRASORO ALGEBRAS
XIAOLI KONG AND CHENGMING BAI
We classify the compatible left-symmetric superalgebraic structures on thesuper-Virasoro algebras satisfying certain natural conditions.
1. Introduction
Left-symmetric algebras (also called pre-Lie algebras, quasiassociative algebras, orVinberg algebras) are a class of natural algebraic systems appearing in many fieldsin mathematics and mathematical physics. They were first mentioned by A. Cayley[1890] as a kind of rooted tree algebra and later arose again from the study ofconvex homogeneous cones [Vinberg 1963], affine manifolds and affine structureson Lie groups [Koszul 1961], and deformation of associative algebras [Gersten-haber 1963]. They play an important role in the study of symplectic and complexstructures on Lie groups and Lie algebras [Andrada and Salamon 2005; Chu 1974;Dardie and Medina 1996b; 1996a; Lichnerowicz and Medina 1988], phase spacesof Lie algebras [Bai 2006; Kupershmidt 1994; 1999a], certain integrable systems[Bordemann 1990; Svinolupov and Sokolov 1994], classical and quantum Yang–Baxter equations [Diatta and Medina 2004; Etingof and Soloviev 1999; Golubchikand Sokolov 2000; Kupershmidt 1999b], combinatorics [Ebrahimi-Fard 2002],quantum field theory [Connes and Kreimer 1998], vertex algebras [Bakalov andKac 2003], and operads [Chapoton and Livernet 2001]. [Burde 2006] gives asurvey.
The super version of left-symmetric algebras, the left-symmetric superalgebras,also appeared in many other fields; see for example [Chapoton and Livernet 2001;Gerstenhaber 1963; Vasil’eva and Mikhalev 1996]. To our knowledge, they werefirst introduced by Gerstenhaber [1963] to study the Hochschild cohomology ofassociative algebras.
On the other hand, the Virasoro and super-Virasoro algebras are not only a classof important infinite-dimensional Lie algebras and Lie superalgebras but are also ofconsiderable interest in physics. For example, they are the fundamental algebraic
MSC2000: 17B68, 17D25, 17B60.Keywords: left-symmetric superalgebra, Virasoro algebra, super-Virasoro algebra.
43
44 XIAOLI KONG AND CHENGMING BAI
structures in conformal and superconformal field theory. As Kupershmidt [1999a]pointed out, a compatible left-symmetric algebra structure on the Virasoro algebracan be regarded as the “nature of the Virasoro algebra”; in fact, on the Virasoroalgebra V given there, such an algebraic structure satisfies
(1-1) cc = xmc = cxm = 0 and xm xn = f (m, n)xm+n +ω(m, n)c,
where f (m, n) and ω(m, n) are two complex-valued functions, and {xm, c | m ∈ Z}
is a basis of the Virasoro algebra V satisfying
(1-2) [c, xn] = 0 and [xm, xn] = (m − n)xm+n +c
12(m3
− m)δm+n,0.
The condition (1-1) is natural since it means that the compatible left-symmetricalgebra is still graded and c is also a central extension given by ω(m, n). Moreover,in [Kong et al. 2007], we proved that any compatible left-symmetric algebraicstructure on the Virasoro algebra V satisfying (1-1) is isomorphic to one of theexamples given in [Kupershmidt 1999a].
In this paper, we study the compatible left-symmetric superalgebraic structureson the super-Virasoro algebras. Motivated by the study in the case of the ordinaryVirasoro algebra, we classify such left-symmetric superalgebras satisfying somenatural conditions like (1-1). The paper is organized as follows. In Section 2,we give some necessary definitions, notations, and basic results on left-symmetricsuperalgebras and the super-Virasoro algebras. We also give the classification ofcompatible left-symmetric algebraic structures on the ordinary Virasoro algebrasatisfying (1-1). In Section 3, we study the compatible left-symmetric superalge-braic structures on the centerless super-Virasoro algebras satisfying certain naturalconditions. In Section 4, we discuss the nontrivial central extensions of the left-symmetric superalgebras obtained in Section 3 whose supercommutator is a super-Virasoro algebra.
Throughout, all algebras are over the complex field C and the indices m, n, l ∈ Z
and r, s, t ∈ Z + θ for θ = 0 or θ = 1/2, unless otherwise stated.
2. Preliminaries and fundamental results
Let (A, ·) be an algebra over a field F. A is said to be a superalgebra if the under-lying vector space of A is Z2-graded, that is, A = A0 ⊕ A1, and Aα · Aβ ⊂ Aα+β ,for α, β ∈ Z2. An element of A0 is called even and an element of A1 is called odd.
Definition 2.1. A Lie superalgebra is a superalgebra A= A0⊕A1 with an operation[ , ] satisfying the conditions
[a, b] = −(−1)αβ[b, a],(2-1)
[a, [b, c]] = [[a, b], c] + (−1)αβ[b, [a, c]](2-2)
SUPERALGEBRAIC STRUCTURES ON THE SUPER-VIRASORO ALGEBRAS 45
where a ∈ Aα, b ∈ Aβ , c ∈ A, and α, β ∈ Z2.
Definition 2.2. A superalgebra A is called a left-symmetric superalgebra if theassociator
(2-3) (x, y, z) := (x · y) · z − x · (y · z)
of A satisfies
(2-4) (x, y, z)= (−1)αβ(y, x, z) for all x ∈ Aα, y ∈ Aβ , z ∈ A, α, β ∈ Z2.
Obviously, if A= A0⊕A1 is a Lie superalgebra or a left-symmetric superalgebra,then A0 is an ordinary Lie algebra or a left-symmetric algebra, respectively. LettingA be a left-symmetric superalgebra, it is easy to show that the supercommutator
[x, y] = x · y − (−1)αβ y · x for all x ∈ Aα, y ∈ Aβ , α, β ∈ Z2(2-5)
defines a Lie superalgebra G(A), which is called the subadjacent Lie superalgebraof A, and A is also called the compatible left-symmetric superalgebraic structureon the Lie superalgebra G(A).
On the other hand, we recall the definition of the super-Virasoro algebras. Thereare two super-Virasoro algebras that correspond to N = 1 [Ramond 1971] andN = 2 [Neveu and Schwarz 1971a; 1971b] superconformal field theory. In fact, letθ = 0 or 1/2, which corresponds to the Ramond case or the Neveu–Schwarz case,respectively. Let SV = SV0 ⊕ SV1 denote a super-Virasoro algebra with a basis{Lm,Gr , c | m ∈ Z, r ∈ Z + θ}. The superbrackets are defined as
(2-6)
[Lm, Ln] = (m − n)Lm+n +c
12(m3
− m)δm+n,0,
[Lm,Gr ] =(m
2− r
)Gm+r ,
[Gr ,Gs] = 2Lr+s +c
12(4r2
− 1)δr+s,0,
[SV0, c] = [SV1, c] = 0,
where the even subspace SV0 is spanned by {Lm, c | m ∈ Z} and the odd subspaceSV1 is spanned by {Gr | r ∈ Z + θ}. Obviously, SV0 is nothing but an ordinaryVirasoro algebra. A class of compatible left-symmetric algebraic structures on theVirasoro algebra satisfying (1-1) were given in [Kupershmidt 1999a]. Such left-symmetric algebras were classified in [Kong et al. 2007].
Theorem 2.3 [Kong et al. 2007]. Any compatible left-symmetric algebraic struc-ture on the Virasoro algebra SV0 satisfying (1-1) is isomorphic to one of the (mu-tually nonisomorphic) left-symmetric algebras given by the multiplication
(2-7) Lm Ln =−n(1+εn)1+ε(m+n)
Lm+n +c
24(m3
− m + (ε− ε−1)m2)δm+n,0
46 XIAOLI KONG AND CHENGMING BAI
for all m, n ∈ Z, where c is an annihilator and Re ε > 0, ε−1 /∈ Z or Re ε = 0,Im ε > 0.
3. Compatible left-symmetric superalgebraic structures on the centerlesssuper-Virasoro algebras
Let SV = SV0⊕SV1 be a centerless super-Virasoro algebra with a basis {Lm,Gr |
m ∈ Z, r ∈ Z+θ}, and let the superbrackets be given by (2-6) with c = 0. Theorem2.3 motivates us to consider the compatible left-symmetric superalgebraic struc-tures on SV that also satisfy the “graded” condition, that is, the multiplications ofthe compatible left-symmetric superalgebraic structures on SV that satisfy
(3-1)Lm · Ln = f (m, n)Lm+n,
Gr · Lm = h(r,m)Gm+r ,
Lm · Gr = g(m, r)Gm+r ,
Gr · Gs = d(r, s)Lr+s,
where f , g, h, and d are C-valued functions. Then the supercommutators give thesuper-Virasoro algebra SV if and only if f (m, n), g(m, r), h(r,m), and d(r, s)satisfy
(3-2)f (m, n)− f (n,m)= m − n,
d(r, s)+ d(s, r)= 2, g(m, r)− h(r,m)= m/2 − r.
Furthermore, the functions f (m, n), g(m, r), h(r,m), and d(r, s) define a left-symmetric superalgebra with a basis {Lm,Gr | m ∈ Z, r ∈ Z + θ} if and only ifthey satisfy the equations
(Lm, Ln, L l)= (−1)0·0(Ln, Lm, L l),
(Lm,Gr , Ln)= (−1)0·1(Gr , Lm, Ln),
(Gr ,Gs, Lm)= (−1)1·1(Gs,Gr , Lm),
(Lm, Ln,Gr )= (−1)0·0(Ln, Lm,Gr ),
(Lm,Gr ,Gs)= (−1)0·1(Gr , Lm,Gs),
(Gr ,Gs,G t)= (−1)1·1(Gs,Gr ,G t).
These equations are equivalent to the equations
(3-3)
(m − n) f (m + n, l)= f (n, l) f (m, n + l)− f (m, l) f (n,m + l),
(m − n)g(m + n, r)= g(n, r)g(m, n + r)− g(m, r)g(n,m + r),
(m/2 − r)h(m + r, n)= h(r, n)g(m, n + r)− f (m, n)h(r,m + n),
(m/2 − r)d(m + r, s)= d(r, s) f (m, r + s)− g(m, s)d(r,m + s),
2 f (r + s,m)= h(s,m)d(r,m + s)+ h(r,m)d(s,m + r),
2g(r + s, t)= d(s, t)h(r, s + t)+ d(r, t)h(s, r + t).
Proposition 3.1. Any compatible left-symmetric superalgebraic structure V onSV satisfies (3-1) if and only if the functions in (3-1) satisfy (3-2) and (3-3).
SUPERALGEBRAIC STRUCTURES ON THE SUPER-VIRASORO ALGEBRAS 47
By Theorem 2.3, we only need to consider the case that
(3-4) f (m, n)=−n(1+εn)1+ε(m+n)
,
where Re ε > 0, ε−1 /∈ Z or Re ε = 0, Im ε > 0.
Theorem 3.2. For a fixed ε satisfying Re ε > 0, ε−1 /∈ Z or Re ε = 0, Im ε > 0and f (m, n) satisfying (3-4), there is exactly one solution satisfying (3-2) and (3-3)given by
(3-5)g(m, r)=
−(m2 + r)(1 + 2εr)
1 + 2ε(m + r),
h(r,m)=−m(1 + εm)
1 + 2ε(m + r), d(r, s)=
1 + 2εs1 + ε(r + s)
,
for m, n ∈Z and r, s ∈Z+θ , which define a compatible left-symmetric superalgebraV ε on SV.
Proof. It is easy to verify that f (m, n) given in (3-4) and g(m, r), h(r,m), andd(r, s) given in (3-5) satisfy (3-2) and (3-3). On the other hand, set
G(m, r)= g(m, r)1+2ε(m+r)1+2εr
,
H(r,m)= h(r,m)1+2ε(m+r)1+εm
, D(r, s)= d(r, s)1+ε(r +s)1+2εs
.
Then we only need to prove that
G(m, r)= −m/2 − r, H(r,m)= −m, D(r, s)= 1.
We rewrite equations (3-2) and (3-3) involving g(m, r), h(r,m), d(r, s) as
G(m, r)(1 + 2εr)− H(r,m)(1 + εm)= (m/2 − r)(1 + 2ε(m + r)),(3-6)
D(r, s)(1 + 2εs)+ D(s, r)(1 + 2εr)= 2 + 2ε(r + s),(3-7)
and
(m − n)G(m + n, r)= G(n, r)G(m, n + r)− G(m, r)G(n,m + r)(3-8)
(m/2 − r)H(m + r, n)= H(r, n)G(m, n + r)+ nH(r,m + n),(3-9)
(m/2 − r)D(m + r, s)= −(r + s)D(r, s)− G(m, s)D(r,m + s),(3-10)
−2m = H(s,m)D(r,m + s)+ H(r,m)D(s,m + r),(3-11)
2G(r + s, t)= D(s, t)H(r, s + t)+ D(r, t)H(s, r + t).(3-12)
Letting r = s in equation (3-7), we have
D(s, s)= 1 for all s ∈ Z + θ.(3-13)
48 XIAOLI KONG AND CHENGMING BAI
In fact, D(r, s) 6= 0 for all r, s ∈ Z + θ . Otherwise, assume there exist r1 and s1
such that D(r1, s1)= 0. Letting r = s = r1 and m = s1 − r1 in (3-11), we have
−(s1 − r1)= H(r1, s1 − r1)D(r1, s1)= 0.
Hence r1 = s1. This contradicts (3-13).Let m = 0 and r = s in (3-10) and (3-6), r = s = −t in (3-12), m = −2s, r = s
in (3-6) and (3-10), and m = −2s 6= 0, r = 3s in (3-10). We know that
(3-14)
G(0, s)= −s,
H(s, 0)= 0,
D(−s, s)= D(3s, s)= 1 for all s ∈ Z + θ.
G(2s,−s)= 0,
H(s,−2s)= 2s,
Letting m = −2(n + r) in (3-9) and m + r + s = 0 in (3-11), we have
(3-15)0 = (n + 2r)H(−2n − r, n)+ nH(r,−n − 2r),
−2m = H(−m − r,m)+ H(r,m).
Letting r = s in (3-11), we have −m = H(r,m)D(r,m + r). So
(3-16) H(r,m)= −m/D(r,m + r).
By (3-13)–(3-16), we have
(3-17)D(−2n − r,−n − r)= D(r,−n − r),
1/D(r,m + r)+ 1/D(−m − r,−r)= 2.
Letting −n − r = s and m + r = s in (3-17), we have D(r, s) = D(2s + r, s) andD(−s,−r) = D(−s,−2s − r) for all r, s ∈ Z + θ . Thus by induction, we knowthat
(3-18) D(r, s)= D(2ks + r, s) and D(−s,−r)= D(−s,−2ks − r)
for all k ∈ Z. Therefore, D(r, r) = D((2k + 1)r, r) = D(r, (2k + 1)r) = 1 for allk ∈ Z. Let r = s in (3-12). Then by (3-16) and (3-18), we have
G(2s, t)= D(s, t)H(s, s + t)= D(s, t) −s−tD(s, 2s+t)
= −s − t.
Letting m = 2s in (3-6), we have
(3-19) H(t, 2s)= −2s for all s, t ∈ Z + θ.
Case I. θ = 1/2. Then D(θ,±θ) = D(±θ, θ) = 1. Hence D(k + θ,±θ) =
D(±θ, k + θ) = 1. That is, D(r,±1/2) = D(±1/2, r) = 1 for all r ∈ Z + θ .Assume that for any |r1| ≤ |s1|, we have D(r1, s1)= 1. Then
D(r1, s1)= D(2ks1 + r1, s1)= 1 and D(s1, r1)= D(s1, 2ks1 + r1)= 1.
SUPERALGEBRAIC STRUCTURES ON THE SUPER-VIRASORO ALGEBRAS 49
For any r ∈ Z + θ , there exist k ∈ Z, r1 ∈ Z + θ , and |r1| ≤ |s1| such that r =
2ks1 +r1. Therefore D(r, s1)= D(s1, r)= 1 for all r ∈ Z+θ . Hence by induction,we know that D(r, s) = 1 for any r, s ∈ Z + θ . Therefore H(r,m) = −m andG(m, r)= −m/2 − r for all m ∈ Z and r ∈ Z + θ .
Case II. θ = 0. Let m = −2t 6= 0 and s = r = 2t in (3-11). Then by (3-19), wehave 2t = H(2t,−2t)D(2t, 0)= 2t D(2t, 0). Therefore
(3-20) D(2t, 0)= 1 and D(0, 2t)= 1 for all t ∈ Z.
Letting r = 0 and m = s 6= 0 in (3-10) and (3-11), we have
(3-21)m/2 = −m D(0,m)− G(m,m),
−2m = H(m,m)+ H(0,m).
So
H(m,m)= −2m − H(0,m)= −2m +m
D(0,m)= −2m −
2m2
m+2G(m,m).
By (3-6), we have H(m,m)= −m and G(m,m)= −3m/2, or
(3-22) H(m,m)=−m
1+εmand G(m,m)=
−3m−4εm2
2+4εm.
In fact, the latter case cannot hold for any m 6= 0. Otherwise, assume that thereexists a nonzero integer m1 satisfying (3-22) with m1 replacing m. Then
D(0,m1)= −12
−G(m1,m1)
m1=
1+εm11+2εm1
.
Letting m = −s = m1 and r = 0 or −m1 in (3-10), we havem12
= m1 D(0,−m1)− G(m1,−m1),
3m12
D(0,−m1)= 2m1 − G(m1,−m1)D(−m1, 0).
Hence (3/2)D(0,−m1)=2−D(0,−m1)D(−m1, 0)+(1/2)D(−m1, 0). By (3-7),we have D(0,−m1)= 1 and D(−m1, 0)= 1, or
D(0,−m1)=3 − εm1
1 − 2εm1and D(−m1, 0)= −1 − εm1.
Since ε 6= 0 and ε−1 /∈ Z, we know that 1/D(0,m1)+ 1/D(−m1, 0) 6= 2, whichcontradicts (3-17). Hence H(m,m) = −m and G(m,m) = −3m/2 for all m ∈ Z.By (3-21) and (3-6), we have H(0,m)= −m and G(m, 0)= −m/2. Letting r = 0and m 6= 0 in (3-8) and (3-9), we have
n2−m2
=−nG(m, n)+mG(n,m), (m/2)H(m, n)=−nG(m, n)−n(m+n).
50 XIAOLI KONG AND CHENGMING BAI
So H(m, n)+2G(n,m)= −2(m +n). By (3-6), (3-13), and (3-16), we know thatH(m, n)= −n and G(n,m)= −n/2 − m and D(m, n)= 1 for all m, n ∈ Z. �
4. Compatible left-symmetric superalgebraic structures on thesuper-Virasoro algebras
We now consider the central extensions of the left-symmetric superalgebras ob-tained in Section 3 whose supercommutator is a super-Virasoro algebra SV.
Let A be a left-symmetric superalgebra, and let ω : A × A → C be a bilinearform. It defines a multiplication on the space A = A ⊕ Cc by the rule
(4-1) (x + λc) · (y +µc)= x · y +ω(x, y)c for all x, y ∈ A, λ, µ ∈ C.
Let
(4-2) B(x, y, z) := ω(x · y, z)−ω(x, y · z).
Then it is easy to show that A is a left-symmetric superalgebra if and only if
(4-3) B(x, y, z)= (−1)αβB(y, x, z)
for all x ∈ Aα, y ∈ Aβ , and z ∈ A, where α, β ∈ Z2. The algebra A is called acentral extension of A. By construction, the bilinear form
(4-4) �(x, y)= ω(x, y)− (−1)αβω(y, x)
for all x ∈ Aα, y ∈ Aβ and z ∈ A, where α, β ∈ Z2, defines a central extension ofthe Lie superalgebra G(A).
Let the left-symmetric superalgebra V ε on a centerless Virasoro algebra SV
be given through Theorem 3.2. Since a super-Virasoro algebra SV is a centralextension of a centerless super-Virasoro algebra SV, it is natural to consider thecentral extension Vε = V ε ⊕ Cc of V ε such that Vε is a compatible left-symmetricsuperalgebraic structure on the super-Virasoro algebra SV with c being the anni-hilator of Vε , that is, the products of Vε are given by
(4-5)
Lm · Ln = f (m, n)Lm+n +ω(Lm, Ln)c,
Lm · Gr = g(m, r)Gm+r +ω(Lm,Gr )c,
Gr · Lm = h(r,m)Gm+r +ω(Gr , Lm)c,
Gr · Gs = d(r, s)Lr+s +ω(Gr ,Gs)c,
c · c = c · Lm = Lm · c = c · Gr = Gr · c = 0,
where the functions f (m, n), g(m, r), h(r,m), and d(r, s) satisfy (3-4) and (3-5).
SUPERALGEBRAIC STRUCTURES ON THE SUPER-VIRASORO ALGEBRAS 51
For convenience, set
(4-6)ω(Lm, Ln)= ϕ(m, n),
ω(Gr , Lm)= ρ(r,m),
ω(Lm,Gr )= ψ(m, r),
ω(Gr ,Gs)= σ(r, s).
So the supercommutators of Vε give a super-Virasoro algebra SV if and only ifϕ(m, n), ψ(m, r), ρ(r,m), and σ(r, s) satisfy
(4-7)
ϕ(m, n)−ϕ(n,m)=112(m
3− m)δm+n,0,
σ (r, s)+ σ(s, r)=112(4r2
− 1)δr+s,0,
ψ(m, r)− ρ(r,m)= 0.
By (4-3), we have
B(Lm, Ln, L l)= (−1)0·0 B(Ln, Lm, L l),
B(Lm, Ln,Gr )= (−1)0·0 B(Ln, Lm,Gr ),
B(Lm,Gr , Ln)= (−1)0·1 B(Gr , Lm, Ln),
B(Lm,Gr ,Gs)= (−1)0·1 B(Gr , Lm,Gs),
B(Gr ,Gs, Lm)= (−1)1·1 B(Gs,Gr , Lm),
B(Gr ,Gs,G t)= (−1)1·1 B(Gs,Gr ,G t).
These are equivalent to the equations
(m − n)ϕ(m + n, l)= f (n, l)ϕ(m, n + l)− f (m, l)ϕ(n,m + l),(4-8)
(m − n)ψ(m + n, r)= g(n, r)ψ(m, n + r)− g(m, r)ψ(n,m + r),(4-9)
(m2 − r)ρ(m + r, n)= h(r, n)ψ(m, n + r)− f (m, n)ρ(r,m + n),(4-10)
(m2 − r)σ (m + r, s)= d(r, s)ϕ(m, r + s)− g(m, s)σ (r,m + s),(4-11)
2ϕ(r + s,m)= h(s,m)σ (r,m + s)+ h(r,m)σ (s,m + r),(4-12)
2ψ(r + s, t)= d(s, t)ρ(r, s + t)+ d(r, t)ρ(s, r + t).(4-13)
Proposition 4.1. Any compatible left-symmetric superalgebraic structure V onSV satisfies (4-5) if and only if the functions in (4-5) satisfy (3-4), (3-5) and (4-7)–(4-13).
If a central extension Vε of V ε given by ω satisfying Equation (4-5) defines acompatible left-symmetric superalgebraic structure on SV, then ϕ(m, n) defines acentral extension of SV0. By Theorem 2.3, we know that
(4-14) ϕ(m, n)=124(m
3− m + (ε− ε−1)m2)δm+n,0.
Theorem 4.2. For a fixed ε ∈ C satisfying Re ε > 0, ε−1 /∈ Z or Re ε= 0, Im ε > 0,suppose the functions f (m, n), g(m, r), h(r,m), and d(r, s) satisfy (3-4) and (3-5),
52 XIAOLI KONG AND CHENGMING BAI
and suppose ϕ(m, n) satisfies (4-14). Then there is exactly one solution satisfying(4-7)–(4-13). It is given by
(4-15)σ(r, s)=
124(4r2
− 1 + 2(ε− ε−1)r)δr+s,0,
φ(m, r)= ρ(r,m)= 0
for m ∈ Z and r, s ∈ Z+θ , and it defines a compatible left-symmetric superalgebraVε on SV.
Proof. It is easy to verify that the ϕ(m, n) given in (4-14) and the σ(r, s), φ(m, r),and ρ(r,m) given in (4-15) satisfy (4-7)–(4-13).
On the other hand, let m = 0 and r + s 6= 0 in (4-11). Then we have
−rσ(r, s)= d(r, s)ϕ(0, r + s)− g(0, s)σ (r, s)= sσ(r, s).
Hence σ(r, s)= 0 for all r +s 6= 0. Letting r = s and m = −2s in Equation (4-12),we have 2ϕ(2s,−2s)= h(s,−2s)σ (s,−s)+h(s,−2s)σ (s,−s)= 4sσ(s,−s). Soσ(s,−s)= (1/24)(4s2
− 1 + 2(ε− ε−1)s). Thus
σ(r, s)=1
24(4r2− 1 + 2(ε− ε−1)r)δr+s,0 for all r, s ∈ Z + θ.
Next, we prove that ψ(m, r)= ρ(r,m)= 0 for all m ∈ Z and r ∈ Z + θ . Thereare two cases.
Case I. θ = 1/2. Letting m = n = 0 in (4-10), we have
−rρ(r, 0)= h(r, 0)ψ(0, r)− f (0, 0)ρ(r, 0)= 0.
So ρ(r, 0) = 0. By (4-7), we know that ψ(0, r) = 0. Letting n = 0 in (4-9),we have mψ(m, r)= ψ(m, r)g(0, r)−ψ(0,m + r)g(m, r)= −rψ(m, r). Hence(m + r)ψ(m, r)= 0. Therefore, we have
ψ(m, r)= ρ(r,m)= 0 for all m ∈ Z, r ∈ Z + 1/2.
Case II. θ = 0. Letting n = 0 and m = −r 6= 0 in (4-10), we have ψ(0, 0) =
ρ(0, 0) = 0. Letting m = n = 0 and r 6= 0 in (4-10) and letting m = r = 0 andn 6= 0 in (4-9), we have
ρ(r, 0)= ψ(0, r)= 0 and ψ(n, 0)= ρ(0, n)= 0 for all r, n ∈ Z, r, n 6= 0.
Let r = 0, m, n 6= 0 in (4-9) and (4-10), we have
ψ(m, n) n1+2εn
−ψ(n,m) m1+2εm
= 0,
m2ψ(n,m)+ψ(m, n)n(1+εn)
1+2εn= 0.
Since ε 6= 0 and ε−1 /∈ Z, we have ψ(n,m)= ρ(n,m)= 0 for all m, n ∈ Z. �
SUPERALGEBRAIC STRUCTURES ON THE SUPER-VIRASORO ALGEBRAS 53
By Theorem 2.3, it is easy to show that the Vε are mutually nonisomorphic forall ε ∈ C satisfying Re ε > 0, ε−1 /∈ Z or Re ε = 0, Im ε > 0. Altogether, fromTheorem 2.3, Proposition 3.1, Theorem 3.2, Proposition 4.1, and Theorem 4.2, wehave the following conclusion.
Theorem 4.3. Any compatible left-symmetric superalgebra on a super-Virasoroalgebra satisfying (4-5) is isomorphic to one of the following (mutually noniso-morphic) left-symmetric superalgebras given by the multiplications
Lm · Ln = −n(1+εn)
1+ε(m+n)Lm+n +
c24(m3
− m + (ε− ε−1)m2)δm+n,0,
Lm · Gr = −(m/2+r)(1+2εr)
1+2ε(m+r)Gm+r ,
Gr · Lm = −m(1+εm)
1+2ε(m+r)Gm+r ,
Gr · Gs =1+2εs
1+ε(r +s)Lr+s +
c24(4r2
− 1 + 2(ε− ε−1)r)δr+s,0,
where m, n ∈ Z, r, s ∈ Z+θ , c is an annihilator, and Re ε >0, ε−1 /∈ Z or Re ε= 0,Im ε > 0.
Acknowledgments
C. Bai thanks Professor B. A. Kupershmidt for important suggestions and encour-agement. This work was supported by the National Natural Science Foundationof China (10571091, 10621101), NKBRPC (2006CB805905), Program for NewCentury Excellent Talents in University.
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Received August 29, 2007. Revised December 14, 2007.
XIAOLI KONG
SCHOOL OF MATHEMATICAL SCIENCES
XIAMEN UNIVERSITY
XIAMEN 361005CHINA
CHENGMING BAI
CHERN INSTITUTE OF MATHEMATICS AND LPMCNANKAI UNIVERSITY
TIANJIN 300071CHINA
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES
ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
We study the space of nearly Kähler structures on compact 6-dimensionalmanifolds. In particular, we prove that the space of infinitesimal defor-mations of a strictly nearly Kähler structure (with scalar curvature scal)modulo the group of diffeomorphisms is isomorphic to the space of prim-itive coclosed (1, 1)-eigenforms of the Laplace operator for the eigenvalue2 scal/5.
1. Introduction
A nearly Kahler manifold is an almost Hermitian manifold (M, g, J ) with theproperty that (∇X J )X = 0 for all tangent vectors X , where ∇ denotes the Levi-Civita connection of g. A nearly Kahler manifold is called strictly nearly Kahler if(∇X J ) is nonzero for every nonzero tangent vector X . Besides Kahler manifolds,there are two main families of examples of compact nearly Kahler manifolds:naturally reductive 3-symmetric spaces, which are classified by A. Gray and J.Wolf [1968]; and twistor spaces over compact quaternion-Kahler manifolds withpositive scalar curvature that are endowed with the nonintegrable canonical almostcomplex structure (see [Nagy 2002]).
A nearly Kahler manifold of dimension 4 is automatically a Kahler surface,and the only known examples of non-Kahler compact nearly Kahler manifolds indimension 6 are the 3-symmetric spaces G2/SU3, SU3/S1
× S1, Sp2/S1× Sp1,
and Sp1 × Sp1 × Sp1/Sp1. Moreover, J.-B. Butruille [2005] has recently shownthat there are no other homogeneous examples in dimension 6.
On the other hand, using previous results of R. Cleyton and A. Swann on G-structures with skew-symmetric intrinsic torsion, Nagy [2002] proved that everycompact simply connected nearly Kahler manifold M is isometric to a Riemannianproduct M1 × · · · × Mk , such that for each i , Mi is a nearly Kahler manifoldbelonging to the following list: Kahler manifolds, naturally reductive 3-symmetricspaces, twistor spaces over compact quaternion-Kahler manifolds with positivescalar curvature, and 6-dimensional nearly Kahler manifolds.
MSC2000: 58E30, 53C10, 53C15.Keywords: infinitesimal deformations, SU3 structures, nearly Kähler manifolds, Gray manifolds.
57
58 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
It is thus natural to concentrate on the 6-dimensional case, especially becausein this dimension non-Kahler nearly Kahler manifolds have several interesting fea-tures: they carry real Killing spinors (and thus are automatically Einstein withpositive scalar curvature) and they are defined by a SU3 structure whose intrinsictorsion is skew-symmetric. These manifolds were intensively studied by A. Grayin the 70’s, thus motivating the following
Definition. A compact strictly nearly Kahler manifold of dimension 6 is called aGray manifold.
Our main goal is to study the deformation problem for Gray manifolds. Weconsider simultaneous deformations of the metric and of the almost complex struc-ture. Indeed M. Verbitsky [2007] proved that on a 6-dimensional almost complexmanifold there is up to constant rescaling at most one strictly nearly Kahler metric.Conversely it is well known (see [Baum et al. 1991] or Section 4 below) that ona manifold (M6, g) that is not locally isometric to the standard sphere, there isat most one compatible almost complex structure J such that (M, g, J ) is nearlyKahler.
We start by studying deformations of SU3 structures and then use the characteri-zation of Gray manifolds as SU3 structures satisfying a certain exterior differentialsystem in order to compute the space of infinitesimal deformations of a givenGray structure modulo diffeomorphisms. In particular, we prove that this spaceis isomorphic to some eigenspace of the Laplace operator acting on 2-forms (seeTheorem 4.1 for a precise statement).
2. Algebraic preliminaries
Let V denote the standard 6-dimensional SU3 representation space, which comesequipped with the Euclidean product g ∈ SymV ∗, the complex structure J ∈
End(V ), the fundamental 2-form ω( · , ·) = g(J · , ·) ∈ 32V ∗, and the complexvolume element ψ+
+ iψ−∈3(3,0)V ∗.
These objects satisfy the compatibility relations ω ∧ψ±= 0 and ψ+
∧ψ−=
(2/3)ω3= 4dv, where dv denotes the volume form of the metric g. It is easy to
check that ψ+ and ψ− are related by
(1) ψ−(X, Y, Z) := −ψ+(J X, Y, Z).
We identify elements of V and V ∗ using the isomorphism induced by g. Forany orthonormal basis {ei } of V adapted to J (that is, J (e2i−1)= e2i ) we have
ω = e12+ e34
+ e56,
ψ+= e135
− e146− e236
− e245, ψ−= e136
+ e145+ e235
− e246.
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES 59
The following formulas are straightforward (it is enough to check them for X = e1
and use the transitivity of the SU3 action on spheres):
ψ+∧ (X yψ+)= X ∧ω2, ψ+
∧ (X yψ−)= −J X ∧ω2,(2)
ψ−∧ (X yψ+)= J X ∧ω2, ψ−
∧ (X yψ−)= X ∧ω2.(3)
Let 3 :3pV →3p−2V denote the metric adjoint of the wedge product with ω,that is, 3=
12
∑i Jeiyeiy. It is easy to check that
(4) 3(X yψ±)= 0 and 3(X ∧ψ±)= J X yψ± for all X ∈ V
and
(5) 3(τ ∧ω)= ω∧3τ + (3 − p)τ for all τ ∈3pV .
We next describe the decomposition into irreducible summands of 32V and33V . We use the notation 3(p,q)+(q,p)V for the projection of 3(p,q)V onto thereal space 3p+q V . Then
32V = (3(1,1)0 V ⊕ Rω)⊕3(2,0)+(0,2)V,
where the first two summands consist of J -invariant and the last of J -antiinvariantforms. Here 3(1,1)0 V is the space of primitive (1, 1)-forms, that is, the kernel ofthe contraction map3. The map ξ 7→ ξyψ+ defines an isomorphism of the secondsummand 3(2,0)+(0,2)V with V . For 3-forms we have the irreducible decomposi-tion
(6) 33V = (31V ∧ω)⊕3(3,0)+(0,3)V ⊕3(2,1)+(1,2)0 V .
The second summand3(3,0)+(0,3)V is 2-dimensional and spanned by the formsψ±.The third summand 3(2,1)+(1,2)0 V is 12-dimensional and can be identified with thespace of symmetric endomorphisms of V anticommuting with J . Because of theSchur lemma, the map given by taking the wedge product with ω vanishes on thelast two summands.
An endomorphism A of V (not necessarily skew-symmetric) acts on p-formsby the formula
(7) (A?u)(X1, . . . , X p) := −
p∑i=1
u(X1, . . . , A(X i ), . . . , X p).
A more convenient way of writing this action is
(8) A?u = −
p∑i=1
A∗(ei )∧ ei y u,
60 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
where A∗ denotes the metric adjoint of A. Taking A = J , we obtain the formspaces 3(p,q)+(q,p)V as eigenspaces of (J?)2 for the eigenvalues −(p − q)2. Forexample, J?ϕ = 0 for any 2-form ϕ ∈3(1,1)V .
Let Sym−V denote the space of symmetric endomorphisms anticommuting withJ . This space is clearly invariant by composition with J . The map S 7→ S?ψ+,with S ∈ Sym−V , defines an isomorphism of SU(3)-representations
Sym−V ∼=3(2,1)+(1,2)0 V,
showing in particular that Sym−V is irreducible. Taking (1) into account, we re-mark that for S ∈ Sym−V we have S?ψ+
= (J S)?ψ−. Notice that tr(S)= 0 for allS ∈ Sym−V .
Let h be any skew-symmetric endomorphism anticommuting with J . Then themap h 7→ g(h · , ·) identifies the space of skew-symmetric endomorphism anticom-muting with J with3(2,0)+(0,2)V . Using the isomorphism ξ 7→ ξyψ+ we can statethis as
Lemma 2.1. An endomorphism F of V anticommuting with J can be written in aunique way F = S +ψ+
ξ for some S ∈ Sym−V and ξ ∈ V , where ψ+
ξ denotes theskew-symmetric endomorphism of V defined by g(ψ+
ξ · , ·)= ψ+(ξ, · , ·).
Corresponding to the decomposition of 33V given in (6), we have
Lemma 2.2. An exterior 3-form u ∈33V can be written in a unique way
u = α∧ω+ λψ++µψ−
+ S?ψ+,
for some α ∈ V , λ,µ ∈ R, and S ∈ Sym−V . Its contraction with ω satisfies
(9) 3u = 2α.
Proof. The contraction map3 obviously vanishes on3(3,0)+(0,3)V ⊕3(2,1)+(1,2)0 V ,
so by (5) we have 3u =3(α∧ω)= 2α. �
The space of symmetric endomorphisms commuting with J is identified with3(1,1)V via the map h 7→ϕ( · , ·) :=g(h J · , ·), which in particular maps the identityof V to the fundamental formω. If ϕ is a (1, 1)-form with corresponding symmetricendomorphism h, then h = h0 +(1/6) tr(h) id, where h0 denotes the trace-free partof h. As a consequence of this formula and Schur’s Lemma we find that
(10) h?ψ+=
16 tr(h) id?ψ+
= −12 tr(h)ψ+
for all symmetric endomorphisms h commuting with J .In the rest of this section we recall several properties and formulas related to the
Hodge ∗-operator, which we will use in later computations. We consider the scalar
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES 61
product 〈 · , ·〉 on 3k V characterized by the fact that the basis
{ei1 ∧ · · · ∧ eik | 1 ≤ i1 < · · ·< ik ≤ 6}
is orthonormal. With respect to this scalar product, the interior and exterior prod-ucts are adjoint operators:
〈X yω, τ 〉 = 〈ω, X ∧ τ 〉 for all X ∈ V , ω ∈3k V , and τ ∈3k−1V .
We define the Hodge star operator ∗ :3k V →36−k V by
ω∧ ∗τ := 〈ω, τ 〉dv for all ω, τ ∈3k V ,
where dv = 1/6ω3 denotes the volume form (dv = e123456 in our notations). It iswell known that the following relations are satisfied:
(11) ∗ω =12ω
2, 〈∗ω, ∗τ 〉 = 〈ω, τ 〉, ∗2= (−1)k on 3k V
From the expression of ψ+ and ψ− in any orthonormal basis {ei } we see that∗ψ+
= ψ− and ∗ψ−= −ψ+. For later use we compute the Hodge operator on
3(2,1)+(1,2)0 V , too. Let S ∈ Sym−V and α ∈ 3
(2,1)+(1,2)0 V . We have α ∧ψ−
= 0,whence
〈α, S?ψ+〉dv
by (8)= −
∑i〈α, S(ei )∧ ei yψ+
〉dv = −∑
i〈ei ∧ S(ei ) yα,ψ
+〉dv
= (S?α)∧ ∗ψ+= (S?α)∧ψ−
= S?(α∧ψ−)−α∧ (S?ψ−)
= −α∧ (S?ψ−)= 〈α, ∗(S?ψ−)〉dv,
where we used that ∗2= −1 on 3-forms to get the last equality. This shows that
(12) ∗(S?ψ−)= S?ψ+ and ∗ (S?ψ+)= −S?ψ−.
There are two other formulas which we will use later. Let ϕ0 be a primitive (1, 1)-form, let ξ ∈ V , and let α ∈3pV then
(13) ∗(ϕ0 ∧ω)= −ϕ0 and ∗ (ξ ∧α)= (−1)pξy ∗α .
3. Deformations of SU3 structures
Let M be a smooth 6-dimensional manifold.
Definition. An SU3 structure on M is a reduction of the frame bundle of M toSU3. It consists of a 5-tuple (g, J, ω,ψ+, ψ−), where g is a Riemannian metric,J is a compatible almost complex structure, ω is the corresponding fundamental2-form ω( · , ·)= g(J · , ·), and ψ+
+iψ− is a complex volume form of type (3, 0).
62 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
If M carries an SU3 structure, each tangent space Tx M has an SU3 representationisomorphic to the standard one, and so all algebraic results of the previous sectiontransfer verbatim to global results on M . In what remains, we will usually identifytangent vectors and 1-forms on M using the isomorphism induced by the metric g.
Let (gt , Jt , ωt , ψ+t , ψ
−t ) be a smooth family of SU3 structures on M . We omit
the index t when the above tensors are evaluated at t = 0, and we use the dot todenote the derivative at t = 0 in the direction of t .
We start with the study of the 1-jet at t = 0 of the family of U3 structures(gt , Jt , ωt).
Lemma 3.1. There exist a vector field ξ , a section S of Sym−M and a section h ofSym+M (that is, symmetric endomorphism commuting with J ) such that
g = g((h + S) · , ·),(14)
J = J S +ψ+
ξ ,(15)
ω = ϕ+ ξ yψ+,(16)
where ϕ is the (1, 1)-form defined by ϕ( · , ·)= g(h J · , ·).
Proof. Let us write gt( · , ·) = g( ft · , ·), so that g( · , ·) = g( f · , ·). We thendenote by h := (1/2)( f − J f J ) and S := (1/2)( f + J f J ) the J -invariant andJ -antiinvariant parts of f , which are clearly g-symmetric endomorphisms. Thisproves the first relation, which actually holds for deformations of almost Hermitianstructures in all dimensions.
Since J 2t = − id T M , we see that J anticommutes with J . Lemma 2.1 then
shows that J = S +ψ+
ξ for some section S of Sym−M and some vector field ξ .Differentiating the relation gt(Jt X, Y )+ gt(X, Jt Y )= 0 yields
0 = g(J X, Y )+ g(X, JY )+ g( J X, Y )+ g(X, J Y )= 2g(S J X, Y )+ 2g(SX, Y ),
thus proving S = −S J = J S. The last formula follows directly from (14) and (15):
ω(X, Y )= g(J X, Y )+ g( J X, Y )= g((h + S)J X, Y )+ g((J S +ψ+
ξ )X, Y )
= g(h J X, Y )+ (ξ yψ+)(X, Y ). �
This result actually says that the tangent space to the set of all U3 structures onM at (g, J, ω) is parametrized by a section (ξ, S, ϕ) of the bundle T M⊕Sym−M⊕
3(1,1)M . We now go forward and describe the 1-jet of a family of SU3 structures.Since the reduction from a U3 structure to a SU3 structure is given by a sectionin some S1-bundle, it is not very surprising that the extra freedom in the tangentspace is measured by a real function (µ in the notation below):
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES 63
Lemma 3.2. The derivatives at t = 0 of ψ+t and ψ−
t are given by
ψ+= −ξ ∧ω+ λψ+
+µψ−−
12 S?ψ+,(17)
ψ−= −Jξ ∧ω−µψ+
+ λψ−−
12 S?ψ−,(18)
where λ= (1/4) tr(h) and µ is some smooth function on M.
Proof. By Lemma 2.2, we can write
(19) ψ+= α∧ω+ λψ+
+µψ−+ Q?ψ
+,
for some functions λ and µ, a 1-form α, and some section Q of Sym−M .The fact thatψ+
t defines — in addition to the U3 structure (gt , Jt)— a SU3 struc-ture is characterized by the two equations
(20) gt(ψ+
t , ψ+
t )= 4 and ψ+
t (Jt X, Y, Z)= ψ+
t (X, Jt Y, Z).
We consider the symmetric endomorphism ft introduced above, which correspondsto gt in the ground metric g. Since the identity acts on 3-forms by −3 id, the firstpart of (20) reads g( ft?ψ
+t , ψ
+t )= −12. Differentiating this at t = 0 and using the
fact that ψ+ and S?ψ+ live in orthogonal components of 33 M , we obtain
0 = g( f ?ψ+, ψ+)− 6g(ψ+, ψ+)= g((h + S)?ψ+, ψ+)− 24λby (10)
= 6 tr(h)− 24λ.
This determines the function λ. We next differentiate the identity ψ+t ∧ωt = 0 at
t = 0. Since the wedge product with ω vanishes on ψ+ and ψ− and on Q?ψ+
∈
3(2,1)+(1,2)0 M , we get
0 = ψ+∧ω+ψ+
∧ ω = α∧ω2+ψ+
∧ (ϕ+ ξ yψ+)by (2)= (α+ ξ)∧ω2,
showing that α = −ξ . Finally, we differentiate the second part of (20) at t = 0:
ψ+(J X, Y, Z)+ψ+( J X, Y, Z)= ψ+(X, JY, Z)+ψ+(X, J Y, Z).
Using (15) and (19), this is equivalent to the expression
−(ξ ∧ω)(J X, Y, Z)+ (Q?ψ+)(J X, Y, Z)+ψ+(J SX, Y, Z)+ψ+(ψ+
ξ X, Y, Z)
being skew-symmetric in X and Y . It is easy to check that
−(ξ ∧ω)(J X, Y, Z)+ψ+(ψ+
ξ X, Y, Z)= (Jξ ∧ω)(X, Y, Z);
therefore the above condition reduces to
(Q?ψ+)(J X, Y, Z)+ψ+(J SX, Y, Z)= (Q?ψ
+)(X, JY, Z)+ψ+(X, J SY, Z).
64 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
Using (7), this last relation becomes
(21) ψ+((2Q + S)J X, Y, Z)=ψ+(X, (2Q + S)JY, Z) for all X, Y, Z ∈ T M .
The set of all elements of the form 2Q + S satisfying this relation is a SU3-invariant subspace of Sym−M . But Sym−M is irreducible, and not every elementof Sym−M satisfies (21) (to see this, just pick any element in Sym−M and checkdirectly). This shows that 2Q + S = 0.
Finally, (18) is a straightforward consequence of (17). We simply differentiatethe formula Jt?ψ
+t = 3ψ−
t (obtained from (1)) at t = 0 and compute. �
Summarizing, we have shown that the tangent space to the set of all SU3 struc-tures on M at (g, J, ω,ψ+, ψ−) is parametrized by a section (ξ, S, ϕ, µ) of thevector bundle T M⊕Sym−M⊕3(1,1)M⊕RM , where RM is the trivial line bundleover M .
Let α :32 M → T M denote the metric adjoint of the linear map
X ∈ T M 7→ X yψ+∈32 M.
A simple check shows that
(22) α(X yψ+)= 2X, α(X yψ−)= −2J X, α(τ )= 0 for all τ ∈3(1,1)M .
Using the map α, we derive a useful relation between the components ψ+ and ωof any infinitesimal SU3 deformation:
(23) 3ψ+ by (9)= −2ξ
by (22)= −α(ω).
4. Deformations of Gray manifolds
Definition. A Gray structure on a 6-dimensional manifold M is a SU3 structureG := (g, J, ω,ψ+, ψ−) that satisfies the exterior differential system
(24)dω = 3ψ+,
dψ−= −2ω∧ω
A Gray manifold is a compact manifold endowed with a Gray structure.
Since SU3 ⊂ Spin6, every Gray manifold is automatically spin. It follows fromthe work of Reyes-Carrion [1993] that a Gray manifold is a strictly nearly Kahler6-dimensional compact manifold with scalar curvature scal = 30. We refer to[Gray 1976] for an introduction to nearly Kahler geometry. We will use lateron the relations ∇Xω = X yψ+ and ∇Xψ
+= −X ∧ ω, which show that ∇ω
and ∇ψ+ are SU3-invariant tensor fields on M . Moreover the second equationimmediately implies that ψ+ and ω are both eigenforms of the Laplace operatorfor the eigenvalue 12.
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES 65
Note let M be a compact 6-dimensional manifold with some Gray structure G onit. We denote by M the connected component of G in the space of Gray structureson M . Let D be the group of diffeomorphisms of M isotopic to the identity. Thisgroup acts on M by pull-back and the orbits of this action form the moduli spaceof deformations of G.
The 1-jet of a curve of Gray structures (gt , Jt , ωt , ψ+t , ψ
−t ) at G becomes at
the infinitesimal level a tuple γ := (g, J , ω, ψ+, ψ−) determined by a section(ξ, S, ϕ, µ) of the bundle T M ⊕ Sym−M ⊕3(1,1)M ⊕ RM via (15)–(18), whichsatisfies the linearized system of (24), that is,
(25)dω = 3ψ+,
dψ− = −4ω∧ω.
The space of all tuples γ is called the virtual tangent space of M at G and isdenoted by TGM. The Lie algebra χ(M) of D maps to TGM by X 7→ L X G. Itsimage, denoted by χG(M), is a vector space isomorphic to χ(M)/K(g), whereK(g) denotes the set of Killing vector fields on M with respect to g. The spaceof infinitesimal Gray deformations of G is, by definition, the vector space quotientTGM/χG(M).
The main purpose of this section is to precisely describe this space.
Theorem 4.1. Let G := (g, J, ω,ψ+, ψ−) be a Gray structure on a manifold Msuch that (M, g) is not the round sphere S6. Then the space of infinitesimal defor-mations of G is isomorphic to the eigenspace for the eigenvalue 12 of the restrictionof the Laplace operator1 to the space of coclosed primitive (1, 1)-forms3(1,1)0 M.
Proof. A simple but very useful remark is that (except on the round sphere S6),a Gray structure is completely determined by its underlying Riemannian metric.The reason is that the metric defines a unique line of Killing spinors with posi-tive Killing constant, which, in turn, defines the almost complex structure, and,together with the exterior derivative of the Kahler form, one recovers the wholeSU3 structure.
We claim that the dependence of the Gray structure on the metric is smooth.Let 6M denote the spin bundle of M . By [Baum et al. 1991, p. 137], there is(up to rescalings) exactly one Killing spinor 9 with Killing constant 1/2, which isobtained as a section of6M , that is parallel with respect to the modified connection∇X := ∇X − (1/2)X · (here “ · ” denotes the Clifford product). Since ∇ dependssmoothly on g, so does 9. The almost complex structure J is then defined bythe equation J X · 9 = i X · 9 (see [Baum et al. 1991, p. 136]), and so J de-pends smoothly on 9. Finally, since the Kahler form ω( · , ·) = g(J · , ·) dependssmoothly on g, so does its exterior derivative dω = ψ+.
66 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
By the Ebin slice theorem [Berger and Ebin 1969], each infinitesimal deforma-tion of G has a unique representative γ = (g, J , ω, ψ+, ψ−) ∈ TGM such that
(26) δg = 0 and trg g = 0.
Let (ξ, S, ϕ, µ) be the section of the bundle T M ⊕ Sym−M ⊕ 3(1,1)M ⊕ RMdetermined by γ via the equations (15)–(18). We have to interpret the system of(25) and (26) in terms of (ξ, S, ϕ, µ).
We start by taking the exterior product with ψ+ in the first equation of (25) anduse (17) to get dω∧ψ+
= 3ψ+∧ψ+
= 3µψ−∧ψ+
=−12µdv.On the other hand,using (16) and taking (3) into account yields dω ∧ψ+
= d(ϕ + ξ yψ+)∧ψ+=
d((ϕ+ ξ yψ+)∧ψ+)= d(ξ ∧ω2), whence
(27) −12µdv = d(ξ ∧ω2).
We apply the contraction 3 to the first equation of (25) and use (9), (16), and (17):
(28) −6ξ = 33ψ+=3dω =3dϕ+3d(ξ yψ+).
To compute the last term, we apply the general formula (23) to the particular de-formation of the SU3 structure defined by the flow of ξ : 3(L ξψ
+)= −α(L ξω).Since dω = 3ψ+ and dψ+
= 0, we get
3d(ξ yψ+)=3(L ξψ+)= −α(L ξω)
= −α(d(ξ yω))−α(ξ y dω)by (22)
= −α(d Jξ)− 6ξ,
which, together with (28), yields
(29) 3(dϕ)= α(d Jξ).
We now examine the second equation of the system (25). From (18) we get
dψ−= −d Jξ ∧ω+ 3Jξ ∧ψ+
− dµ∧ψ++ dλ∧ψ−
− 2λω2−
12 d(S?ψ−).
We apply the contraction 3 to this formula and use the second equation of (25)together with (4) and (5):
(30) −4ω =3(−4ω∧ω)+ 4(3ω)ω =3dψ−+ 4(3ω)ω
= −d Jξ −3(d Jξ)ω− 3ξ yψ+− Jdµ yψ+
+ Jdλ yψ−
− 8λω−123d(S?ψ−)+ 4(3ω)ω.
Applying α to this equality and using (1), (16) and (23) yields
−8ξ = −α(d Jξ)− 6ξ − 2Jdµ+ 2dλ−12α3d(S?ψ−).
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES 67
From (29) we then get
(31) ξ = Jdµ+123dϕ− dλ+
14α3d(S?ψ−).
Lemma 4.2. Let (g, J, ω,ψ+, ψ−) be a Gray structure on a manifold M. Forevery section S of Sym−M and (1, 1)-form ϕ, the following relations hold:
3dϕ = δh + 2dλ(32)
δh = −Jδϕ(33)
δ(S?ψ+)= −3d(S?ψ−)− 2δS yψ+(34)
α3d(S?ψ−)= −2δS(35)
3δ(S?ψ+)= 0(36)
where h is the endomorphism defined by ϕ( · , ·)= g(h J · , ·) and λ= (1/4) tr(h)=(1/2)3ϕ. In the above formulas, δ stands for the usual codifferential when appliedto an exterior form but stands for the divergence operator (see [Besse 1987, 1.59])when acting on symmetric tensors.
Since the proof is rather technical, we postpone it to the end of this section.Using (14), (32), (33) and (35), the relation (31) becomes
(37) ξ = Jdµ+12δh −
12δS = Jdµ− Jδϕ−
12δg.
From (26) and (37) we obtain
(38) ξ = Jdµ− Jδϕ.
On the other hand, (38) shows that µ is an eigenfunction of 1 with eigenvalue 6:
(39) 1µ= δdµ= δ(Jξ)= − ∗ d ∗ (Jξ)= −12 ∗ d(ξ ∧ω2)
by (27)= 6µ.
Now, the Obata theorem (see [Obata 1962, Theorem 3]) says that on a compactn-dimensional Einstein manifold of positive scalar curvature scal, every eigenvalueof the Laplace operator is greater than or equal to scal/(n − 1), and equality canonly occur on the standard sphere. Since (M, g) is not isometric to the standardsphere and is Einstein with scalar curvature scal = 30, the Obata theorem thusimplies that µ = 0. Since λ = (1/4) tr(h) = (1/4) tr(g), the second part of (26)also shows that λ= 0. Taking (26), (33), (34), and (38) into account, Equation (30)now becomes
−4ϕ− 4ξ yψ+= −4ω = −d Jξ −3(d Jξ)ω− 3ξ yψ+
−123d(S?ψ−)
= −d Jξ −3(d Jξ)ω− 3ξ yψ++
12δ(S?ψ
+)− ξ yψ+,
whenced Jξ = −3(d Jξ)ω+ 4ϕ+
12δ(S?ψ
+).
68 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
Applying 3 to this relation and using (36) yields 3(d Jξ) = 33(d Jξ), and sod Jξ = 4ϕ +
12δ(S?ψ
+). By (38) we have Jξ = δϕ, and thus 1(Jξ) = δd Jξ =
4δϕ = 4Jξ , that is, Jξ is an eigenform of the Laplace operator with eigenvalue 4.On the other hand, the Bochner formula on 1-forms,
1= ∇∗∇ + Ric = ∇
∗∇ + 5 id
shows, by integration over M , that 4 cannot be an eigenvalue of 1; so ξ mustvanish identically. From Lemmas 3.1 and 3.2, we get
ψ+= −
12 S?ψ+, ψ−
= −12 S?ψ−, ω = ϕ ∈�
(1,1)0 M.
Plugging these equations into (25) yields dϕ = −32 S?ψ+ and
δ(S?ψ+)= − ∗ d ∗ (S?ψ+)by (12)
= ∗d(S?ψ−)= −2 ∗ dψ−
by (25)= 8 ∗ (ω∧ω)
by (13)= −8ϕ.
Thus ϕ is a coclosed eigenform of the Laplace operator for the eigenvalue 12.Conversely let us assume that ϕ ∈�
(1,1)0 M is coclosed and satisfies 1ϕ = 12ϕ.
We have to show that ϕ defines an infinitesimal deformation of the given Graystructure. The main point is to remark that dϕ is a form in �(2,1)+(1,2)0 M .
Lemma 4.3. If ϕ is a coclosed form in �(1,1)0 M , then dϕ ∈�(2,1)+(1,2)0 M.
Proof. Using Lemma 2.2, this amounts to checking that dϕ satisfies the system
(40)
dϕ ∧ψ+= 0,
dϕ ∧ψ−= 0,
〈dϕ, X ∧ω〉 = 0 for all X ∈ T M .
Each of these equations follows easily:
dϕ ∧ψ+= d(ϕ ∧ψ+)= 0,
dϕ ∧ψ−= d(ϕ ∧ψ−)−ϕ ∧ dψ− by (24)
= 2ϕ ∧ω2 by (11)= 4〈ϕ, ω〉 = 0,
〈dϕ, X ∧ω〉 = 〈3dϕ, X〉by (32)
= −〈Jδϕ, X〉dv = 0. �
Consequently, there is a unique section S of Sym−M with dϕ = −(3/2)S?ψ+.Taking ξ =0, λ=0, andµ=0, the equations (15)–(18) define an infinitesimal SU3-deformation by ω := ϕ, ψ+
:= −(1/2)S?ψ+, and ψ−:= −(1/2)S?ψ−. It remains
to show that (ω, ψ+, ψ−) satisfy the linearized system (25). The first equation isclear by definition, and the second is equivalent to d(S?ψ−)= 8ϕ∧ω. Using again(12) and (13), this last equation is equivalent to δ(S?ψ+) = −8ϕ, which followsdirectly from the hypothesis on ϕ and the definition of S. �
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES 69
Proof of Lemma 4.2. By the isomorphism ξ 7→ ξyψ+, we have ∇X J = ψ+
X ∈
3(0,2)+(2,0)M , and so 〈ϕ,∇X J 〉 = 0 for all vectors X . Identifying ϕ with thecorresponding skew-symmetric endomorphism of T M , we compute in a local or-thonormal basis {ei } parallel at some point:
3dϕ =12 Jei y ei y (ek ∧ ∇ekϕ)=
12 Jei y ∇eiϕ−
12 Jei y (ek ∧ (∇ekϕ)ei )
=12(∇eiϕ)Jei −
12(∇Jeiϕ)ei +
12 ek(∇ekϕ)(ei , Jei )
= (∇eiϕ)Jei + d〈ϕ, ω〉 − ekϕ(ei , (∇ek J )ei )
= (∇ei (ϕ J ))ei −ϕ((∇ei J )ei )+ 2dλ− 2ek〈ϕ,∇ek J 〉
= −(∇ei h)(ei )+ 2dλ= δh + 2dλ.
This proves (32). To prove the second relation, we notice that
g((∇ei J )(ϕei ), X)= ψ+(ei , ϕei , X)= 2〈ψ+
X , ϕ〉 = 0,
and so
δh = −(∇ei h)ei = (∇ei (Jϕ))ei = −Jδϕ+ (∇ei J )ϕei = −Jδϕ.
We will now use several times (and signal this by a star above the equality sign inthe calculations below) the fact that any SU3-equivariant map Sym−M →32 M isautomatically zero by the Schur lemma. For every 3-form τ we can write
3d(τ )=12 Jei y ei y (ek ∧ ∇ekτ)= ek ∧
( 12 Jei y ei y ∇ekτ
)+ Jek y ∇ekτ
= d3τ −12 ek ∧ (∇ek J )ei y ei y τ + Jek y ∇ekτ.
In particular, for τ = S?ψ+ we have 3τ = 0 and ek ∧ (∇ek J )ei y ei y τ = 0, and sofrom (8) and the remark above we get
3d(S?ψ−)= Jek y ∇ek (S?ψ−)= Jek y ((∇ek S)?ψ−)+ Jek y (S?(∇ekψ
−))
∗= −Jek y ((∇ek S)e j ∧ e j yψ−)
= −〈(∇ek S)e j , Jek〉e j yψ−+ (∇ek S)e j ∧ (Jek y e j yψ−)
∗= 〈(∇ek S)Je j , ek〉Je j yψ+
+ (∇ek S)e j ∧ (Jek y e j yψ−)
= −δS yψ++ (∇ek S)e j ∧ (Jek y e j yψ−).
So by (1)
(41) 3d(S?ψ−)= −δS yψ++ (∇ek S)e j ∧ (ek y e j yψ+).
70 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
On the other hand, we have
δ(S?ψ+) = ek y ∇ek (Se j ∧ e j yψ+)∗= ek y ((∇ek S)e j ∧ e j yψ+)
= − δS yψ+− (∇ek S)e j ∧ ek y e j yψ+
by (41)= − 2δS yψ+
−3d(S?ψ−),
thus proving (34). Let σ := (∇ek S)e j ∧ (ek y e j yψ+) denote the last summand in(41). A similar calculation easily shows that J?σ = 0, so σ belongs to 3(1,1)M .From (22) we thus get α3d(S?ψ−)= −α(δS yψ+)= −2δS. Finally, the relation(36) can be checked in the same way:
3δ(S?ψ+)= −12 Jek y ek y (e j y ∇e j (S?ψ
+))
= −12
(e j y ∇e j (Jek y ek y (S?ψ+))− e j y (∇e j J )ek y ek y (S?ψ+)
)∗= δ3(S?ψ+)= 0. �
5. Concluding remarks
So far we have identified the space of infinitesimal deformations of a given Graystructure with the space of coclosed primitive (1, 1)-forms which are eigenformsof the Laplace operator for the eigenvalue 12. To proceed further there are twoimmediate options. One could try to compute the second derivative of a curve ofGray structures and obtain additional equations. However, this leads to quadraticexpressions, which for the moment seem difficult to handle.
A second natural task is to consider the known homogeneous examples and tostudy the question of whether or not there exist at least infinitesimal deformations.This amounts to studying the Laplace operator on 2-forms on certain homogeneousspaces and should reduce to a tractable algebraic problem.
Since a nearly Kahler deformation always gives rise to an Einstein deformation,one could equally well ask for the existence of infinitesimal Einstein deformationson nearly Kahler manifolds. This issue was recently treated in [Moroianu andSemmelmann 2007].
The deformation problem for the standard nearly Kahler structure on S6 has tobe considered separately since the almost complex structure is no longer uniquelydefined. However, since the round metric on S6 has no Einstein deformations,the problem is much simpler in this case. Th. Friedrich [2006] showed that theaction of the isometry group SO7 on the set of nearly Kahler structures on theround sphere S6 is transitive. The isotropy group of this action at the standardnearly Kahler structure is easily seen to be the group G2 (the stabilizer in SO7
of a vector cross product). The space of nearly Kahler structures on the roundsphere is thus isomorphic to SO7/G2 ' RP7; see also [Butruille 2005, Prop. 7.2].More geometrically, the set of nearly Kahler structures compatible with the round
DEFORMATIONS OF NEARLY KÄHLER STRUCTURES 71
metric on S6 can be identified with the set of nonzero real Killing spinors (moduloconstant rescalings), and it is well known that the space of real Killing spinors onS6 is isomorphic to R8.
The counterpart of Theorem 4.1 on S6 can be stated as follows:
Theorem 5.1. Let G := (g, J, ω,ψ+, ψ−) be a Gray structure on S6 such that g isthe round metric. Then the space of infinitesimal deformations of G is isomorphicto the eigenspace for the eigenvalue 6 of the Laplace operator 1 on functions, andthat space is, in particular, 7-dimensional.
Proof. Since there are no Einstein deformations on the round sphere, we mayassume g = 0. From (14) we get ϕ = 0, h = 0, S = 0, and in particular λ= 0 too.Then Equation (38) gives ξ = Jdµ and (39) shows that µ is an eigenfunction of 1on S6 corresponding to its first nonzero eigenvalue, 6. These eigenfunctions (alsocalled first spherical harmonics) satisfy ∇X dµ = −µX for all tangent vectors X .We define the infinitesimal SU3 deformation ω := ξyψ+, ψ+
:= −ξ ∧ω+µψ−
and ψ−:= −Jξ ∧ω−µψ+. A short calculation easily shows that this indeed is
a solution of the linearized system (25). �
References
[Baum et al. 1991] H. Baum, Th. Friedrich, R. Grunewald, and I. Kath, Twistors and Killing spinorson Riemannian manifolds, Teubner-Texte zur Mathematik 124, B. G. Teubner, Stuttgart, 1991.MR 94a:53077
[Berger and Ebin 1969] M. Berger and D. Ebin, “Some decompositions of the space of symmetrictensors on a Riemannian manifold”, J. Differential Geometry 3 (1969), 379–392. MR 42 #993Zbl 0194.53103
[Besse 1987] A. L. Besse, Einstein manifolds, vol. 10, Ergebnisse der Mathematik, Springer, Berlin,1987. MR 88f:53087 Zbl 0613.53001
[Butruille 2005] J.-B. Butruille, “Classification des variétés approximativement kähleriennes ho-mogènes”, Ann. Global Anal. Geom. 27:3 (2005), 201–225. MR 2006f:53060 Zbl 1079.53044
[Friedrich 2006] Th. Friedrich, “Nearly Kähler and nearly parallel G2-structures on spheres”, Arch.Math. (Brno) 42 (2006), 241–243. MR 2322410
[Gray 1976] A. Gray, “The structure of nearly Kähler manifolds”, Math. Ann. 223:3 (1976), 233–248. MR 54 #6010 Zbl 0345.53019
[Moroianu and Semmelmann 2007] A. Moroianu and U. Semmelmann, “Infinitesimal Einstein De-formations of Nearly Kähler Metrics”, Preprint, 2007. arXiv math/0702455v1
[Nagy 2002] P.-A. Nagy, “Nearly Kähler geometry and Riemannian foliations”, Asian J. Math. 6:3(2002), 481–504. MR 2003m:53043 Zbl 1041.53021
[Obata 1962] M. Obata, “Certain conditions for a Riemannian manifold to be iosometric with asphere”, J. Math. Soc. Japan 14 (1962), 333–340. MR 25 #5479 Zbl 0115.39302
[Reyes-Carrión 1993] R. Reyes-Carrión, Some special geometries defined by Lie groups, PhD thesis,Oxford, 1993.
[Verbitsky 2007] M. Verbitsky, “Hodge theory on nearly Kähler manifolds”, Preprint, 2007. arXivmath/0510618v5
72 ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
[Wolf and Gray 1968] J. A. Wolf and A. Gray, “Homogeneous spaces defined by Lie group auto-morphisms. I”, J. Differential Geometry 2 (1968), 77–114. MR 38 #4625a Zbl 0169.24103
Received June 7, 2007. Revised September 10, 2007.
ANDREI MOROIANU
CENTRE DE MATHEMATIQUES
ECOLE POLYTECHNIQUE
91128 PALAISEAU CEDEX
FRANCE
[email protected]://www.math.polytechnique.fr/cmat/moroianu/moroianu.html
PAUL-ANDI NAGY
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF AUCKLAND
PRIVATE BAG 92019, AUCKLAND MAIL CENTRE
AUCKLAND 1142NEW ZEALAND
[email protected]://www.math.auckland.ac.nz/wiki/Paul Andi Nagy
UWE SEMMELMANN
MATHEMATISCHES INSTITUT
UNIVERSITAT ZU KOLN
WEYERTAL 86-90D-50931 KOLN
GERMANY
[email protected]://www.mi.uni-koeln.de/~semmelma/
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
INFINITE-DIMENSIONAL SANDWICH PAIRS
MARTIN SCHECHTER
For many equations arising in the physical sciences, the solutions are criticalpoints of functionals. This has led to interest in finding critical points ofsuch functionals. If a functional G is semibounded, one can find Palais–Smale (PS) sequences G(uk)→ a and G′(uk)→ 0. These sequences producecritical points if they have convergent subsequences (that is, if G satisfies thePS condition). However, there is no clear method of finding critical pointsof functionals that are not semibounded. In this paper we find pairs of setshaving the property that functionals bounded from below on one set andbounded from above on the other have PS sequences. We can allow bothsets to be infinite-dimensional if we make a slight additional smoothnessrequirement on the functional. This allows us to solve systems of equationsthat could not be solved before.
1. Introduction
Many problems arising in science and engineering call for the solving of the Eulerequations of functionals, that is, equations of the form
G ′(u)= 0,
where G(u) is a C1 functional (usually with physical dimension of energy) arisingfrom the given data. As an illustration, the equation −1u(x) = f (x, u(x)) is theEuler equation of the functional
G(u)=12‖∇u‖
2−
∫F(x, u(x)) dx
on an appropriate space, where
F(x, t)=
∫ t
0f (x, s) ds,
and the norm is that of L2. Solving the Euler equations is tantamount to findingcritical points of the corresponding functional. The classical approach was to look
MSC2000: 35J65, 58E05, 49J35.Keywords: critical point theory, linking, variational methods, saddle point theory, sandwich pairs,
semilinear partial differential equations, critical sequences.
73
74 MARTIN SCHECHTER
for maxima or minima. If one is looking for a minimum, it is not sufficient to knowthat the functional is bounded from below, as is easily observed. If it is boundedfrom below, one can obtain a minimizing sequence satisfying
G(uk)→ a = inf G.
If such a sequence has a convergent subsequence, then we indeed obtain a mini-mum. However, in dealing with such sequences it is difficult, in general, to estab-lish the convergence of a subsequence.
Luckily, there is some help. One can show that there is a sequence, called aPalais–Smale (PS) sequence satisfying
G(uk)→ a and G ′(uk)→ 0,
where a = inf G. It is much easier to establish the existence of a convergent sub-sequence of a PS sequence than of a minimizing sequence. In fact, a minimizingsequence may not have a convergent subsequence while a PS sequence for the samefunctional does. If every PS sequence for G has a convergent subsequence, thenwe say that G satisfies the PS condition.
However, when the functional is not semibounded, there is no clear way ofobtaining critical points. Is there anything that can be used to replace semibound-edness? Expressed otherwise, how can one level the playing field? One approachis called linking. As a substitute for semiboundedness, one can look for suitablesets that separate the given functional, that is, suitable subsets A and B of the spaceE satisfying
(1) a0 := supA
G ≤ b0 := infB
G
for a given C1 functional G on E . There are pairs of subsets such that (1) producesa PS sequence
(2) G(uk)→ a and G ′(uk)→ 0,
where a ≥ b0. If A and B are such that (1) always implies (2), we say that A linksB. Consequently, if A links B and G is a C1 functional on E that satisfies (1) andthe PS condition, then G has a critical point satisfying
G(u)= a ≥ b0 and G ′(u)= 0.
Linking sets exist and are described in the literature; see for example, [Schechter1999].
In [Schechter 2008] we discussed the situation in which one cannot find linkingsets that separate the functional, that is, that satisfy (1). Are there sets such that the
INFINITE-DIMENSIONAL SANDWICH PAIRS 75
opposite of (1) will imply (2)? More precisely, are there sets A and B such that
(3) −∞< b0 := infB
G and a0 := supA
G <∞
implies that there is a sequence satisfying
(4) G(uk)→ c for b0 ≤ c ≤ a0 and G ′(uk)→ 0?
This was answered in the affirmative. Such pairs exist. This has led to
Definition 1. We say that a pair of subsets A and B of a Banach space E forms asandwich, if for any G ∈ C1(E,R) the inequality (3) implies the existence of a PSsequence (4).
The root of this approach comes from the work of Silva [1991] and the author[1993; 1992]. They proved
Theorem 2. Let N be a closed subspace of a Hilbert space E , and let M = N⊥.Assume that at least one of the subspaces M and N is finite-dimensional. Let G bea C1-functional on E such that
m0 := infw∈M
G(w) 6= −∞ and m1 := supv∈N
G(v) 6= ∞.
Then there are a constant c ∈ R and a sequence {uk} ⊂ E such that
G(uk)→ c with m0 ≤ c ≤ m1 and G ′(uk)→ 0.
This theorem, called the “sandwich theorem”, is very useful in dealing withequations or systems for which the corresponding functional is semibounded inone of the directions only on a subspace of finite dimension. However, there aremany systems for which this is not the case. On the other hand, the theorem isprobably not true if both subspaces are infinite-dimensional.
Here, we shall show that the theorem is indeed true if we require a bit morethan mere continuous differentiability of the functional. The requirement we havechosen is present in many applications.
Definition 3. Let E be a Banach space. We shall call a functional G ∈ C1(E,R)
weak-to-weak continuously differentiable if for each sequence
uk → u weakly in E
there exists a renamed subsequence such that
(5) G ′(uk)→ G ′(u) weakly.
For such functionals we have
76 MARTIN SCHECHTER
Theorem 4. Let N be a closed subspace of a Hilbert space E and let M = N⊥.Let G be a weak-to-weak continuously differentiable functional on E such that
m0 := infw∈M
G(w) 6= −∞ and m1 := supv∈N
G(v) 6= ∞.
Then there are a constant c ∈ R and a sequence {uk} ⊂ E such that
G(uk)→ c with m0 ≤ c ≤ m1 and G ′(uk)→ 0.
We shall prove Theorem 4 in Section 2, where we introduce weak sandwich pairs.Applications will be given in Section 3. One purpose of our investigation is tosolve systems of equations of the form
Av = f (x, v, w) and Bw = g(x, v, w),
where A and B are linear partial differential operators. Such systems have manyapplications. For instance, they can describe multiple chemical reactions or stablestates of dynamical systems determined by reaction diffusion equations.
Unlike linking, the order of a sandwich pair is immaterial, that is, if the pair A, Bforms a sandwich, so does B, A. Moreover, we allow sets forming a sandwich pairto intersect. (A description of sandwich pairs can be found in [Schechter 2008].)It follows from Theorem 2 that M and N form a sandwich pair if one of themis finite-dimensional. (Note that m0 ≤ m1.) This is a severe drawback in manyapplications.
The purpose of the present paper is find a counterpart of sandwich pairs thatdeals with the case when both sets in the pair are infinite-dimensional. To do thiswe require weak-to-weak continuous differentiability of the functional as we didin Theorem 4. We call such pairs weak sandwich pairs.
2. Weak sandwich pairs
We now introduce the corresponding definition for the case when both sets A andB are infinite-dimensional.
Definition 5. We shall say that a pair of subsets A and B of a Banach space Eforms a weak sandwich pair, if for any weak-to-weak continuously differentiableG ∈ C1(E,R) the inequality
(6) −∞< b0 := infB
G ≤ a0 := supA
G <∞
implies that there is a sequence {uk} satisfying
(7) G(uk)→ c with b0 ≤ c ≤ a0 and G ′(uk)→ 0.
INFINITE-DIMENSIONAL SANDWICH PAIRS 77
Theorem 6. Let E be a separable Hilbert space, let N be a closed subspace of E ,and let p be any point of N . Let F be a Lipschitz continuous map of E onto Nsuch that F |N = I and ‖F(g)− F(h)‖ ≤ K‖g − h‖ for g, h ∈ E. Suppose alsothat for each finite-dimensional subspace S of E containing p such that F S 6= {0},there is a finite-dimensional subspace S0 6= {0} of N containing p such that v ∈ S0
and w ∈ S implies F(v+w) ∈ S0. (The stipulation that S0 6= {0} is made in casep = 0.) Then A = N and B = F−1(p) form a weak sandwich pair.
Proof. Assume that the theorem is false. Let G be a weak-to-weak continuouslydifferentiable functional on E satisfying (6), where A and B are the subsets of Especified in the theorem, such that there is no sequence satisfying (7). Then thereis a positive number δ > 0 such that
(8) ‖G ′(u)‖ ≥ 2δ
whenever u belongs to the set E = {∈ E : b0 − 2δ ≤ G(u)≤ a0 + 2δ}. Since E isseparable, we can norm it with a norm |u|w satisfying |u|w ≤ ‖u‖ for u ∈ E andsuch that the topology induced by this norm is equivalent to the weak topology ofE on bounded subsets of E .
This can be done as follows. Let {ek} be an orthonormal basis for E . We set
|u|2w =
∞∑k=1
|(u, ek)|2
k2 .
We denote E equipped with this norm by E . For u ∈ E , let h(u)= G ′(u)/‖G ′(u)‖.Then by (8)
(9) (G ′(u), h(u))≥ 2δ for u ∈ E .
LetT = (a0 − b0 + 4δ)/δ,
R = sup�
‖u‖ + T,
BR = {u ∈ E : ‖u‖< R},
B = B R ∩ E,
where� is a bounded open subset of N containing the point p such that ρ(∂�, p)>K T + δ and ρ is the distance in E . For each u ∈ B there is an E neighborhoodW (u) of u such that (G ′(v), h(u))>δ for v ∈ W (u)∩ B. For otherwise there wouldbe a sequence {vk} ⊂ B such that
(10) |vk − u|w → 0 and (G ′(vk), h(u))≤ δ.
Since B is bounded in E , vk → u weakly in E and (5) implies that
(G ′(vk), h(u))→ (G ′(u), h(u))≤ δ
78 MARTIN SCHECHTER
in view of (10). This contradicts (9). Let B be the set B with the inherited topologyof E . It is a metric space, and W (u) ∩ B is an open set in this space. Thus,{W (u) ∩ B} for u ∈ B is an open covering of the paracompact space B (see forexample [Kelley 1955]). Consequently, there is a locally finite refinement {Wτ } ofthis cover. For each τ there is an element uτ such that Wτ ⊂ W (uτ ). Let {ψτ }
be a partition of unity subordinate to this covering. Each ψτ is locally Lipschitzcontinuous with respect to the norm |u|w and consequently with respect to thenorm of E . Let Y (u) =
∑ψτ (u)h(uτ ) for u ∈ B. Then Y (u) is locally Lipschitz
continuous with respect to both norms. Moreover,
(11) ‖Y (u)‖ ≤∑ψτ (u)‖h(uτ )‖ ≤ 1
and
(12) (G ′(u), Y (u))=∑ψτ (u)(G ′(u), h(uτ ))≥ δ for u ∈ B.
For u ∈�∩ E , let σ(t)u be the solution of
(13) σ ′(t)= −Y (σ (t)) for t ≥ 0 and σ(0)= u.
Note that σ(t)u will exist as long as σ(t)u is in B. Also, it is continuous in (u, t)with respect to both topologies.
Next we note that if u ∈�∩ E we cannot have σ(t)u ∈ B and G(σ (t)u)> b0−δ
for 0 ≤ t ≤ T . For by (13) and (12),
dG(σ (t)u)/dt = (G ′(σ ), σ ′)= −(G ′(σ ), Y (σ ))≤ −δ
as long as σ(t)u ∈ B. Hence if σ(t)u ∈ B for 0 ≤ t ≤ T , we would have
G(σ (T )u)− G(u)≤ −δT = −(a0 − b0 + 4δ).
Thus, we would have G(σ (T )u) < b0 − 4δ. On the other hand, σ(s)u exists for0 ≤ s < T . To see this note that
u − σ(t)u = zt(u) :=
∫ t
0Y (σ (s)u)ds.
By (11), we have ‖zt(u)‖ ≤ t . Consequently, ‖σ(t)u‖ ≤ ‖u‖ + t < R. Thusσ(t)u ∈ B. We can now conclude that for each u ∈�∩ E there is a t ≥ 0 such thatσ(s)u exists for 0 ≤ s ≤ t and G(σ (t)u)≤ b0 − δ. Let
Tu := inf{t ≥ 0 : G(σ (t)u)≤ b0 − δ} for u ∈�∩ E .
Then σ(t)u exists for 0 ≤ t ≤ Tu and Tu < T , and Tu is continuous in u. Define
σ (t)u =
{σ(t)u if 0 ≤ t ≤ Tu ,σ(Tu)u if Tu ≤ t ≤ T ,
for u ∈�∩ E .
INFINITE-DIMENSIONAL SANDWICH PAIRS 79
For u ∈�\ E , define σ (t)u = u for 0 ≤ t ≤ T . Then σ (t)u is continuous in (u, t),and
(14) G(σ (T )u)≤ b0 − δ for u ∈�.
Let
(15) ϕ(v, t)= F σ (t)v for v ∈� and 0 ≤ t ≤ T .
Then ϕ is a continuous map of �× [0, T ] to N . Let
K = {(u, t) : u = σ (t)v for v ∈ Q and t ∈ [0, T ]}.
Then K is a compact subset of E × R. To see this, let (uk, tk) be any sequence inK . Then uk = σ(tk)vk , where vk ∈ Q. Since Q is bounded, there is a subsequencesuch that vk → v0 weakly in E and tk → t0 in [0, T ]. Since Q is convex andbounded, v0 is in Q and |vk − v0|w → 0. Since σ (t) is continuous in E × R,
uk = σ (tk)vk ⇀ σ(t0)v0 = u0 ∈ K .
Each u0 ∈ B has a neighborhood W (u0) in E and a finite-dimensional subspaceS(u0) such that Y (u) ⊂ S(u0) for u ∈ W (u0) ∩ B. Since σ (t)u is continuous in(u, t), for each (u0, t0) ∈ K there is a neighborhood W (u0, t0) ⊂ E × R and afinite-dimensional subspace S(u0, t0) ⊂ E such that zt(u) ⊂ S(u0, t0) for (u, t) ∈
W (u0, t0), where
zt(u) := u − σ (t)u =
{∫ t0 Y (σ (s)u)ds if u ∈ E ,
0 if u 6∈ E .
Since K is compact, there is a finite number of points (u j , t j ) ⊂ K such thatK ⊂ W =
⋃W (u j , t j ). Let S be a finite-dimensional subspace of E containing
p and all the S(u j , t j ) and such that F S 6= {0}. Then for each v ∈ �, we havezt(v) ∈ S. Then by hypothesis, there is a finite-dimensional subspace S0 6= {0} ofN containing p such that F(v− zt(v))∈ S0 for all v ∈�∩ S0. We note that ϕ(u, t)maps �∩ S0 × [0, T ] into S0. For t in [0, T ], let ϕt(v)= ϕ(v, t). Then
(16) ϕt(v) 6= p for v ∈ ∂(�∩ S0)= ∂�∩ S0 and 0 ≤ t ≤ T .
To see this, note that if v ∈ ∂�, then ‖v− p‖ ≤ ‖v− F σ (t)v‖ + ‖F σ (t)v− p‖.Hence ‖F σ (t)v− p‖> K T + δ− t K > 0 for v ∈ ∂� and ≤ t ≤ T , since
‖F σ (t)v− v‖ ≤ K∫ t
0‖σ ′(s)v‖ ds ≤ K t.
80 MARTIN SCHECHTER
Thus (16) holds. Consequently the Brouwer degree d(ϕt , � ∩ S0, p) is defined.Since ϕt is continuous, we have
d(ϕT , �∩ S0, p)= d(ϕ0, �∩ S0, p)= d(I, �∩ S0, p)= 1.
Hence there is a v∈� such that F σ (T )v= p. Consequently σ (T )v∈ F−1(p)= B.In view of (6), this implies G(σ (T )v) ≥ b0, contradicting (14). Thus (7) holds,and the proof is complete. �
Proof of Theorem 4. We take A = N , B = M , p = 0, and F = PN , the projectiononto N . If S is a finite-dimensional subspace such that F S 6= {0}, we take S0 = F S.All of the hypotheses of Theorem 6 are satisfied. �
Definition 7. Let E and F be Banach spaces. We shall call a map J ∈ C(E, F)weak-to-weak continuous if for each sequence
uk → u weakly in E
there exists a renamed subsequence such that
J (uk)→ J (u) weakly in F .
Proposition 8. If A and B is a weak sandwich pair and J is a weak-to-weak con-tinuous diffeomorphism on the entire space having a derivative J ′(u) dependingcompactly on u and satisfying
(17) ‖J ′(u)−1‖ ≤ C for u ∈ E,
then J A and J B is a weak sandwich pair.
Proof. Let G be a weak-to-weak continuously differentiable functional on E sat-isfying
−∞< b0 := infJ B
G ≤ a0 := supJ A
G <∞.
Let G1(u) = G(Ju) for u ∈ E . Then (G1(u), h) = (G ′(Ju), J ′(u)h). If uk → uweakly, then there is a renamed subsequence such that J (uk)→ J (u) weakly andJ ′(uk)→ J ′(u). Hence, (G1(uk), h)→ (G ′(Ju), J ′(u)h), and G1 is weak-to-weakcontinuously differentiable. Moreover,
−∞< b0 := infJ B
G = infJu∈J B
G(Ju)= infB
G1
≤ a0 := supJ A
G = supJu∈J A
G(Ju)= supA
G1 <∞.
Since A and B form a weak sandwich pair, there is a sequence {hk} ⊂ E such that
G1(hk)→ c for b0 ≤ c ≤ a0 and G ′
1(hk)→ 0.
INFINITE-DIMENSIONAL SANDWICH PAIRS 81
If we set uk = J hk , this becomes G(uk)→ c for b0 ≤ c ≤ a0 and G ′(uk)J ′(hk)→ 0.In view of (17), this implies G ′(uk)→ 0. Thus J A and J B form a weak sandwichpair. �
Proposition 9. Let N be a closed subspace of a Hilbert space E with complementM ′
= M ⊕ {v0}, where v0 is an element in E having unit norm, and let δ be anypositive number. Let ϕ(t) ∈ C1(R) be such that 0 ≤ ϕ(t) ≤ 1, ϕ(0) = 1, andϕ(t)= 0 for |t | ≥ 1. Let
F(v+w+ sv0)= v+ [s + δ− δϕ(‖w‖2/δ2)]v0 for v ∈ N , w ∈ M, s ∈ R.
Then A = N ′= N ⊕ {v0} and B = F−1(δv0) form a weak sandwich pair.
Proof. Define
J (v+w+ sv0)= v+w+[s − δ+ δϕ(‖w‖2/δ2)]v0 for v ∈ N , w ∈ M, s ∈ R.
Then J is a diffeomorphism on E satisfying the hypotheses of Proposition 8. Also,A = J N ′ and B = J [M + δv0]. Since N ′ and M + δv0 form a weak sandwich pairby Theorem 4, A and B also form a weak sandwich pair (Proposition 8). �
3. Applications
Let A and B be positive, self-adjoint operators on L2(�) with compact resolvents,where �⊂ Rn . Let F(x, v, w) be a Caratheodory function on �× R2 such that
f (x, v, w)= ∂F/∂v and g(x, v, w)= ∂F/∂w
are also Caratheodory functions satisfying
(18) | f (x, v, w)| + |g(x, v, w)| ≤ C0(|v| + |w| + 1) for v,w ∈ R
and
(19)f (x, t y, t z)/t → α+(x)v+
−α−(x)v−+β+(x)w+
−β−(x)w−,
g(x, t y, t z)/t → γ+(x)v+− γ−(x)v−
+ δ+(x)w+− δ−(x)w−
as t → +∞, y → v, and z → w, where a±= max(±a, 0). We wish to solve the
system
(20)Av = − f (x, v, w),
Bw = g(x, v, w).
Such systems have been studied in the literature by many authors (for example,[Costa 1994; Costa and Magalhaes 1994; de Figueiredo and Felmer 1994; Furtadoand Silva 2001; Furtado et al. 2002a; 2002b; Li and Yang 2004; Schechter 1998;Schechter and Zou 2003; Silva 2001; Tintarev 1999; Peihao et al. 2002; Zou 2001]
82 MARTIN SCHECHTER
and the literature quoted in them). Let λ0(µ0) be the lowest eigenvalue of A(B).We assume that the only solution of
(21)−Av = α+v
+−α−v
−+β+w
+−β−w
−,
Bw = γ+v+
− γ−v−
+ δ+w+
− δ−w−
is v = w = 0.The equations (21) take the place of the equation characterizing the Fucık spec-
trum for a problem involving only one function. Essentially, our hypotheses requirethat (α+, α−, β+, β−, γ+, γ−, δ+, δ−) is not in the “Fucık ” spectrum of (−A,B).
Our first result is
Theorem 10. Assume
2F(x, s, 0)≥ −λ0s2− W1(x) for x ∈� and t ∈ R,(22)
2F(x, 0, t)≤ µ0t2+ W2(x) for x ∈� and t ∈ R,(23)
where Wi (x) ∈ L1(�). Then the system (20) has a solution.
Proof. Let D = D(A1/2)× D(B1/2). Then D becomes a Hilbert space with normgiven by ‖u‖
2D = (Av, v)+ (Bw,w) for u = (v,w) ∈ D. We define
G(u)= b(w)− a(v)− 2∫�
F(x, v, w) dx for u ∈ D,
where a(v)= (Av, v) and b(w)= (Bw,w). Then G ∈ C1(D,R) and
(24) (G ′(u), h)/2 = b(w, h2)− a(v, h1)− ( f (u), h1)− (g(u), h2),
where we write f (u) and g(u) in place of f (x, v, w) and g(x, v, w), respectively.It is readily seen that the system (20) is equivalent to
(25) G ′(u)= 0.
We let N be the set of those (v, 0) ∈ D, and let M be the set of those (0, w) ∈ D.Then M and N are orthogonal closed subspaces such that D = M⊕N . If we defineLu = 2(−v,w) for u = (v,w) ∈ D, then L is a selfadjoint bounded operator onD. Also G ′(u) = Lu + c0(u), where c0(u) = −(A−1 f (u),B−1g(u)) is compacton D. This follows from (18) and the fact that A and B have compact resolvents.It also follows that G ′ has weak-to-weak continuity. For if uk → u weakly, thenLuk → Lu weakly and c0(uk) has a convergent subsequence. Now by (23)
G(0, w)≥ b(w)−µ0‖w‖2−
∫�
W2(x) dx for (0, w) ∈ M.
ThusinfM
G ≥ −
∫�
W (x) dx ≡ b0.
INFINITE-DIMENSIONAL SANDWICH PAIRS 83
On the other hand, (22) implies
G(v, 0)≤ −a(v)+ λ0‖v‖2+
∫�
W1(x) dx for (v, 0) ∈ N .
Thus
(26) supN
G ≤
∫�
W1(x) dx ≡ a0.
We can now apply Theorem 4 to conclude that there is a sequence {uk} ⊂ D suchthat (7) holds. Let uk = (vk, wk). I claim that
(27) ρ2k = a(vk)+ b(wk)≤ C.
For assume that ρk →∞, and let uk = uk/ρk . Then there is a renamed subsequencesuch that uk → u weakly in D, strongly in L2(�) and almost everywhere in �. Ifh = (h1, h2) ∈ D, then
(G ′(uk), h)/ρk = 2b(wk, h2)− 2a(vk, h1)− 2( f (uk), h1)/ρk − 2(g(uk), h2)/ρk .
Taking the limit and applying (18) and (19), we see that u = (v, w) is a solutionof (21). Hence u = 0 by hypothesis. On the other hand, since a(vk)+ b(wk)= 1,there is a renamed subsequence such that a(vk)→ a and b(wk)→ b with a+b = 1.Thus by (19) and (24)
(G ′(uk), (vk, 0))/2ρk = −a(vk)− ( f (uk), vk)/ρk
→ −a −
∫�
(α+v+
−α−v−
+β+w+
−β−w−)v dx,
(G ′(uk), (0, wk))/2ρk = b(wk)− (g(uk), wk)/ρk
→ b −
∫�
(γ+v+
− γ−v−
+ δ+w+
− δ−w−)w dx .
Thus by (7),
a = −
∫�
(α+v+
−α−v−
+β+w+
−β−w−)vdx,
b =
∫�
(γ+v+
− γ−v−
+ δ+w+
− δ−w−)w dx .
Since one of the two numbers a and b is not zero, we cannot have u ≡ 0. Thiscontradiction proves (27). This known, we can use the usual procedures to showthat there is a renamed subsequence such that uk → u in D, and u satisfies (25). �
Theorem 11. In addition, assume that the eigenfunctions of λ0 andµ0 are boundedand 6= 0 almost everywhere in�, and there is a q > 2 such that ‖w‖
2q ≤ Cb(w) for
w ∈ M. Assume 2F(x, 0, t)≤ µ(x)t2 for x ∈� and t ∈ R, where
84 MARTIN SCHECHTER
µ(x)≤ µ0 and µ(x) 6≡ µ0 for x ∈�,(28)
2F(x, s, t)≤ µ0t2− λ0s2 for |t | + |s| ≤ δ,(29)
for some δ > 0. Then the system (20) has a nontrivial solution.
Proof. Let N ′ be the orthogonal complement of N0 = {(ϕ0, 0)} in N , where ϕ0 isthe eigenfunction of A corresponding to λ0. Then N = N ′
⊕ N0. Let M0 be thesubspace of M spanned by the eigenfunctions {(0, ψ)} of B corresponding to µ0,and let M ′ be its orthogonal complement in M . Since N0 and M0 are contained inL∞(�), there is a positive constant ρ such that
(30)a(y)≤ ρ2
⇒ ‖y‖∞ ≤ δ/4 for y ∈ N0,
b(h)≤ ρ2⇒ ‖h‖∞ ≤ δ/4 for h ∈ M0,
where δ is the number in (29). If a(y)≤ ρ2, b(w)≤ ρ2, and |y(x)| + |w(x)| ≥ δ,we write w = h +w′, h ∈ M0, w′
∈ M ′, and
δ ≤ |y(x)| + |w(x)| ≤ |y(x)| + |h(x)| + |w′(x)| ≤ (δ/2)+ |w′(x)|.
Thus
(31)
|y(x)| + |h(x)| ≤ δ/2 ≤ |w′(x)|,
|y(x)| + |w(x)| ≤ 2|w′(x)|.
Now by (29) and (31)
G(y, w)= b(w)− a(y)− 2∫�
F(x, y, w) dx
≥ b(w)− a(y)−∫
|y|+|w|<δ
(µ0w2− λ0 y2)dx − c0
∫|y|+|w|>δ
(|y| + |w| + 1)2 dx
≥ b(w)− a(y)−µ0‖w‖2+ λ0‖y‖
2− c1
∫2|w′|>δ
|w′|qdx
≥ b(w′)−µ0‖w′‖
2− c2b(w′)q/2
≥
(1 −
µ0µ1
− c2b(w′)(q/2)−1)
b(w′) for a(y)≤ ρ2 and b(w)≤ ρ2,
where µ1 is the next eigenvalue of B after µ0. If we reduce ρ accordingly, we canfind a positive constant ν such that
(32) G(y, w)≥ νb(w′), a(y)≤ ρ2, b(w)≤ ρ2.
I claim that either (20) has a nontrivial solution or there is an ε > 0 such that
(33) G(y, w)≥ ε and a(y)+ b(w)= ρ2.
INFINITE-DIMENSIONAL SANDWICH PAIRS 85
For suppose (33) did not hold. Then there would be a sequence {yk, wk} such thata(yk)+b(wk)= ρ2 and G(yk, wk)→ 0. If we write wk =w′
k +hk , w′
k ∈ M ′, andhk ∈ M0, then (32) tells us that b(w′
k)→ 0. Thus a(yk)+ b(hk)→ ρ2. Since N0
and M0 are finite-dimensional, there is a renamed subsequence such that yk → yin N0 and hk → h in M0. By (30), ‖y‖∞ ≤ δ/4 and ‖h‖∞ ≤ δ/4. Consequently(29) implies
(34) 2F(x, y, h)≤ µ0h2− λ0 y2.
Since
G(y, h)= b(h)− a(y)− 2∫�
F(x, y, h)dx = 0,
we have ∫�
(2F(x, y, h)+ λ0 y2−µ0h2)dx = 0.
In view of (34), this implies 2F(x, y, h)≡µ0h2−λ0 y2. For ζ ∈ C∞
0 (�) and t > 0small, we have
2(F(x, y + tζ, h)− F(x, y, h))/t ≤ −λ0((y + tζ )2 − y2)/t.
Taking t → 0, we have f (x, y, h)ζ ≤ −λ0 yζ . Since this is true for all ζ ∈ C∞
0 (�),we have
(35) f (x, y, h)= −λ0 y = −Ay.
Similarly,
2[F(x, y, h + tζ )− F(x, y, h)]/t ≤ µ0[(h + tζ )2 − h2]/t,
and consequently g(x, y, h)ζ ≤ µ0hζ and
(36) g(x, y, h)= µ0h = Bh.
We see from (35) and (36) that (20) has a nontrivial solution. Thus, we may assumethat (33) holds.
Next, we note that there is an ε > 0 depending on ρ such that G(0, w) ≥ ε
for b(w) ≥ ρ > 0. To see this, suppose that {wk} ⊂ M is a sequence such thatG(0, wk)→0 for b(wk)≥ρ. If bk =b(wk)≤C , this implies b(wk)−µ0‖wk‖
2→0
and∫[µ0 −µ(x)]w2
k dx → 0, since
G(0, w)≥ b(w)−µ0‖w‖2+
∫[µ0 −µ(x)]w2dx for w ∈ M.
If we write wk =w′
k +hk and w′
k ∈ M ′ and hk ∈ M0 as before, then this tells us thatb(w′
k)→ 0. Since M0 is finite-dimensional, there is a renamed subsequence suchthat hk → h. But the two conclusions above tell us that h = 0. Since b(h)≥ ρ, we
86 MARTIN SCHECHTER
see that ε > 0 exists for any constant C . If the sequence {bk} is not bounded, wetake wk = wk/
√bk . Then
G(0, wk)/bk ≥ b(wk)−µ0‖wk‖2+
∫[µ0 −µ(x)](wk)
2dx .
Next we note that there is a ν > 0 such that
(37) G(0, w)≥ νb(w) for w ∈ M.
Assuming this for the moment, we see that infB G ≥ ε1 > 0, where
B ={w ∈ M : b(w)≥ ρ2}⋃{
u = (sϕ0, w) : s ≥ 0, w ∈ M, ‖u‖D = ρ},
and ε1 = min{ε, νρ2}. By (26) there is an R > ρ such that supA G = a0 < ∞,
where A = N . By Proposition 9, A and B form a weak sandwich pair. Moreover,G satisfies (6) with ε1 ≤ b0. Hence, there is a sequence {uk} ⊂ D such that (7)holds with c ≥ ε1. Arguing as in the proof of Theorem 10, we see that there is au ∈ D such that G(u)= c ≥ ε1 > 0 and G ′(u)= 0. Since c 6= 0 and G(0)= 0, wesee that u 6= 0, and we have a nontrivial solution of the system (20).
It therefore remains only to prove (37). Clearly ν ≥ 0. If ν = 0, then thereis a sequence {wk} ⊂ M such that G(0, wk) → 0 for b(wk) = 1. Thus there isa renamed subsequence such that wk → w weakly in M , strongly in L2(�) andalmost everywhere in �. Consequently∫
�
[µ0 −µ(x)]w2k dx ≤ 1 −
∫�
µ(x)w2k dx ≤ G(0, wk)→ 0
and
1 =
∫�
µ(x)w2dx ≤ µ0‖w‖2≤ b(w)≤ 1,
which means that we have equality throughout. It follows that w must belong toE(µ0), the eigenspace of µ0. Since w 6≡ 0, we have w 6= 0 almost everywhere.But
∫�[µ0 −µ(x)]w2dx = 0 implies that the integrand vanishes identically on �,
and consequently µ(x) ≡ µ0, violating (28). This establishes (37) and completesthe proof of the theorem. �
References
[Costa 1994] D. G. Costa, “On a class of elliptic systems in RN ”, Electron. J. Differential Equations07 (1994). MR 95e:35065 Zbl 0809.35020
[Costa and Magalhães 1994] D. G. Costa and C. A. Magalhães, “A variational approach to sub-quadratic perturbations of elliptic systems”, J. Diff. Equations 111 (1994), 103–122. MR 95f:35082Zbl 0803.35052
[de Figueiredo and Felmer 1994] D. G. de Figueiredo and P. L. Felmer, “On superquadratic ellipticsystems”, Trans. Amer. Math. Soc. 343:1 (1994), 99–116. MR 94g:35072 Zbl 0799.35063
INFINITE-DIMENSIONAL SANDWICH PAIRS 87
[Furtado and Silva 2001] M. F. Furtado and E. A. B. Silva, “Double resonant problems which arelocally non-quadratic at infinity”, pp. 155–171 in Proceedings of the USA-Chile Workshop on Non-linear Analysis (Viña del Mar, 2000), edited by R. Manasevich and P. Rabinowitz, Electron. J.Differ. Equ. Conf. 6, 2001. MR 2002g:35079 Zbl 0963.35060
[Furtado et al. 2002a] M. F. Furtado, L. A. Maia, and E. A. B. Silva, “On a double resonantproblem in RN ”, Differential Integral Equations 15:11 (2002), 1335–1344. MR 2003g:35064Zbl 1034.35024
[Furtado et al. 2002b] M. F. Furtado, L. A. Maia, and E. A. B. Silva, “Solutions for a resonantelliptic system with coupling in RN ”, Comm. Partial Differential Equations 27:7-8 (2002), 1515–1536. MR 2003f:35076 Zbl 1016.35022
[Kelley 1955] J. L. Kelley, General topology, D. Van Nostrand, Toronto–New York–London, 1955.MR 16,1136c Zbl 0066.16604
[Li and Yang 2004] G. Li and J. Yang, “Asymptotically linear elliptic systems”, Comm. PartialDifferential Equations 29:5-6 (2004), 925–954. MR 2005i:35071 Zbl 02130256
[Peihao et al. 2002] Z. Peihao, Z. Wujie, and Z. Chengkui, “The existence of three nontrivial so-lutions of a class of elliptic systems”, Nonlinear Anal. 49:3 (2002), 431–443. MR 2003b:35057Zbl 1030.35047
[Schechter 1992] M. Schechter, “A generalization of the saddle point method with applications”,Ann. Polon. Math. 57:3 (1992), 269–281. MR 94c:58028 Zbl 0780.35001
[Schechter 1993] M. Schechter, “New saddle point theorems”, pp. 213–219 in Generalized func-tions and their applications (Varanasi, 1991), edited by R. S. Pathak, Plenum, New York, 1993.MR 94i:58034 Zbl 0846.46027
[Schechter 1998] M. Schechter, “Infinite-dimensional linking”, Duke Math. J. 94:3 (1998), 573–595. MR 99h:58034 Zbl 0953.58009
[Schechter 1999] M. Schechter, Linking methods in critical point theory, Birkhäuser, Boston, 1999.MR 2001f:58032 Zbl 0915.35001
[Schechter 2008] M. Schechter, “Sandwich pairs in critical point theory”, Trans. Amer. Math. Soc.(2008). To appear.
[Schechter and Zou 2003] M. Schechter and W. Zou, “Weak linking”, Nonlinear Anal. 55:6 (2003),695–706. MR 2005d:58021 Zbl 1036.58010
[Silva 1991] E. A. d. B. e. Silva, “Linking theorems and applications to semilinear elliptic problemsat resonance”, Nonlinear Anal. 16:5 (1991), 455–477. MR 92d:35108 Zbl 0731.35042
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Received May 8, 2007. Revised November 19, 2007.
MARTIN SCHECHTER
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CALIFORNIA AT IRVINE
IRVINE, CA 92697-3875UNITED STATES
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
ON ALGEBRAICALLY INTEGRABLE OUTER BILLIARDS
SERGE TABACHNIKOV
We prove that if the outer billiard map around a plane oval is algebraicallyintegrable in a certain nondegenerate sense then the oval is an ellipse.
In this note, an outer billiard table is a compact convex domain in the planebounded by an oval (closed smooth strictly convex curve) C . Pick a point x outsideof C . There are two tangent lines from x to C ; choose one of them, say, the rightone from the viewpoint of x , and reflect x in the tangency point. One obtains a newpoint, y, and the transformation T : x 7→ y is the outer (also called dual) billiardmap. We refer to [Tabachnikov and Dogru 2005; Tabachnikov 1995; 2005] forsurveys of outer billiards.
If C is an ellipse then the map T possesses a 1-parameter family of invariantcurves, the homothetic ellipses; these invariant curves foliate the exterior of C .Conjecturally, if an outer neighborhood of an oval C is foliated by the invariantcurves of the outer billiard map, then C is an ellipse — this is an outer version ofthe famous Birkhoff conjecture concerning the conventional, inner billiards.
In this note we show that ellipses are rigid in a much more restrictive sense ofalgebraically integrable outer billiards; see [Bolotin 1990] for the case of innerbilliards.
We make the following assumptions. Let f (x, y) be a (nonhomogeneous) realpolynomial that has zero as a nonsingular value and such that C is a componentof its zero level curve. Thus f is the defining polynomial of the curve C , and ifa polynomial vanishes on C , then it is a multiple of f (see for example [Clemens2003; Fischer 2001; Walker 1978]). Assume that a neighborhood of C is foliatedby invariant curves of the outer billiard map T and that this foliation is algebraicin that its leaves are components of the level curves of a real polynomial F(x, y).Since C itself is an invariant curve, we assume that F(x, y)= 0 on C and that d Fis not identically zero on C . Thus F(x, y) = g(x, y) f (x, y) where g(x, y) is apolynomial not identically zero on C . Under these assumptions, our result is asfollows.
Theorem 1. C is an ellipse.
MSC2000: primary 37J30; secondary 37E40.Keywords: outer billiards, dual billiards, integrability, Birkhoff conjecture.The author was partially supported by an NSF grant DMS-0555803.
89
90 SERGE TABACHNIKOV
Proof. Consider the tangent vector field v = Fy ∂/∂x − Fx ∂/∂y (the symplecticgradient) along C . This vector field is nonzero (except at possibly a finite numberof points) and tangent to C . The tangent line to C at point (x, y) is given by(x +εFy, y −εFx), and the condition that F is T -invariant means that the function
(1) F(x + εFy, y − εFx)
is even in ε for all (x, y) ∈ C . Expand in a series in ε; the first order term in εvanishes automatically and the first nontrivial condition is cubic in ε:
(2) W (F) := Fxxx F3y − 3Fxxy F2
y Fx + 3Fxyy Fy F2x − Fyyy F3
x = 0
on C . We claim that this already implies that C is an ellipse. The idea is thatotherwise the complex curve f = 0 would have an inflection point, in contradictionwith identity (2).
Consider the polynomial
H(F)= det(
Fy −Fx
Fyy Fx − Fxy Fy Fxx Fy − Fxy Fx
).
Lemma 2. (i) v(H(F))= W (F).
(ii) H(F)= H(g f )= g3 H( f ) on C.
(iii) If C ′ is a nonsingular algebraic curve with a defining polynomial g(x, y), thenH(g)(x, y)= 0 if and only if (x, y) is an inflection point of C ′.
Proof. The first two claims follow from straightforward computations. To provethe third, note that H(g) is the second order term in ε of the Taylor expansion ofthe function g(x + εgy, y − εgx); see (1). Hence H(g)= 0 at the points where thetangent line is second order tangent to the curve, that is, at the inflection points. �
It follows from Lemma 2 and (2) that H(F)= const on C . Indeed, v(H(F))=W (F) = 0; hence the directional derivative of H(F) along C is zero. Since Cis convex, H(F) 6= 0. Indeed, if H(F) = 0 then, by Lemma 2, H( f ) = 0 andall points of C are its inflections. Thus we may assume that H(F) = 1 on C . Itfollows that g3 H( f )− 1 vanishes on C and hence
(3) g3 H( f )− 1 = h f,
where h(x, y) is some polynomial.Now consider the situation in CP2. We use the notation C ′ for the complex
algebraic curve given by the homogenized polynomial f (x : y : z)= f (x/z, y/z).Unless C is a conic, this curve has inflection points (not necessarily real). Let dbe the degree of C ′.
Lemma 3. Not all the inflections of C ′ lie on the line at infinity.
ON ALGEBRAICALLY INTEGRABLE OUTER BILLIARDS 91
Proof. Consider the Hessian curve given by
det
fxx fxy fxz
fyx fyy fyz
fzx fzy fzz
= 0.
The intersection points of the curve C ′ with its Hessian curve are the inflectionpoints of C ′ (recall that C ′ is nonsingular). The degree of the Hessian curve is 3(d−
2) because taking two derivatives lowers the degree by 2 and taking the determinantmultiplies terms in threes. By Bezout’s theorem, the total number of inflections,counted with multiplicities, is 3d(d − 2). Also, the order of intersection equalsthe order of the respective inflection and does not exceed d − 2, see for example[Walker 1978]. The number of intersection points of C ′ with a line equals d . Hencethe inflection points of C ′ that lie on a fixed line contribute, at most, d(d − 2) tothe total of 3d(d − 2). The remaining inflection points lie off this line. �
To conclude the proof of Theorem 1, consider a finite inflection point of C ′.According to Lemma 2, at such a point we have f = H( f )= 0, which contradicts(3). This is proves that C is a conic. �
Remarks. First, it would be interesting to remove the nondegeneracy assumptionsin Theorem 1.
Second, a more general version of Birkhoff’s integrability conjecture is as fol-lows. Let C be a plane oval whose outer neighborhood is foliated by closed curves.For a tangent line ` to C , the intersections with the leaves of the foliation definea local involution σ on `. Assume that, for every tangent line, the involution σ isprojective. Conjecturally, then C is an ellipse and the foliation consists of ellipsesthat form a pencil (that is, share four — real or complex — common points). For apencil of conics, the respective involutions are projective: this is a Desargues the-orem; see [Berger 1987]. It would be interesting to establish an algebraic versionof this conjecture.
Acknowledgments
Many thanks to D. Genin for numerous stimulating conversations, to S. Bolotin forcomments on his work [Bolotin 1990], to V. Kharlamov for providing a proof ofLemma 3, to R. Schwartz for interest and criticism, and to the referee for helpfulsuggestions.
References
[Berger 1987] M. Berger, Geometry. I, Universitext, Springer, Berlin, 1987. Translated from theFrench by M. Cole and S. Levy. MR 88a:51001a Zbl 0606.51001
92 SERGE TABACHNIKOV
[Bolotin 1990] S. V. Bolotin, “Integrable Birkhoff billiards”, Vestnik Moskov. Univ. Ser. I Mat.Mekh. 2 (1990), 33–36, 105. In Russian; translated in Mosc. Univ. Mech. Bull. 45:2 (1990), 10–13. MR 91e:58143 Zbl 0708.58015
[Clemens 2003] C. H. Clemens, A scrapbook of complex curve theory, Second ed., Graduate Studiesin Mathematics 55, American Mathematical Society, Providence, RI, 2003. MR 2003m:14001Zbl 1030.14010
[Fischer 2001] G. Fischer, Plane algebraic curves, Student Mathematical Library 15, AmericanMathematical Society, Providence, RI, 2001. MR 2002g:14042 Zbl 0971.14026
[Tabachnikov 1995] S. Tabachnikov, “Billiards”, Panor. Synth. 1 (1995), vi+142. MR 96c:58134Zbl 0833.58001
[Tabachnikov 2005] S. Tabachnikov, Geometry and billiards, Student Mathematical Library 30,American Mathematical Society, Providence, RI, 2005. MR 2006h:51001 Zbl 1119.37001
[Tabachnikov and Dogru 2005] S. Tabachnikov and F. Dogru, “Dual billiards”, Math. Intelligencer27:4 (2005), 18–25. MR 2006i:37121 Zbl 1088.37014
[Walker 1978] R. J. Walker, Algebraic curves, Springer, New York, 1978. MR 80c:14001 Zbl 0399.14016
Received August 1, 2007. Revised August 24, 2007.
SERGE TABACHNIKOV
DEPARTMENT OF MATHEMATICS
PENNSYLVANIA STATE UNIVERSITY
UNIVERSITY PARK, PA 16802UNITED STATES
[email protected]/tabachni/
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
BORNOLOGICAL QUANTUM GROUPS
CHRISTIAN VOIGT
We introduce and study the concept of a bornological quantum group. Thisgeneralizes the theory of algebraic quantum groups in the sense of van Daelefrom the algebraic setting to the framework of bornological vector spaces.Working with bornological vector spaces allows to extend the scope of thelatter theory considerably. In particular, the bornological theory coverssmooth convolution algebras of arbitrary locally compact groups and theirduals. Another source of examples arises from deformation quantization inthe sense of Rieffel. Apart from describing these examples we obtain somegeneral results on bornological quantum groups. In particular, we constructthe dual of a bornological quantum group and prove the Pontrjagin dualitytheorem.
1. Introduction
The concept of a multiplier Hopf algebra introduced by van Daele [1994] extendsthe notion of a Hopf algebra to the setting of nonunital algebras. An importantdifference to the situation for ordinary Hopf algebras is that the comultiplicationof a multiplier Hopf algebra H takes values in the multiplier algebra M(H ⊗ H)and not in H ⊗ H itself. Due to the occurrence of multipliers, certain constructionswith Hopf algebras have to be carried out more carefully in this context. Still,every multiplier Hopf algebra is equipped with a counit and an antipode satisfyinganalogues of the usual axioms. A basic example of a multiplier Hopf algebra isthe algebra Cc(0) of compactly supported functions on a discrete group 0. Thismultiplier Hopf algebra is an ordinary Hopf algebra if and only if the group 0 isfinite.
Algebraic quantum groups form a special class of multiplier Hopf algebras withparticularly nice properties. Every algebraic quantum group admits a dual quantumgroup and the analogue of the Pontrjagin duality theorem holds [van Daele 1998].For instance, the multiplier Hopf algebra Cc(0) associated to a discrete group0 is in fact an algebraic quantum group, its Pontrjagin dual being the complexgroup ring C0. More generally, all discrete and all compact quantum groups can
MSC2000: 16W30, 81R50.Keywords: multiplier Hopf algebras, quantum groups, bornological vector spaces.
93
94 CHRISTIAN VOIGT
be viewed as algebraic quantum groups. In addition, the class of algebraic quan-tum groups is closed under some natural operations including the construction ofthe Drinfeld double [Drabant and van Daele 2001]. Moreover, algebraic quantumgroups give rise to examples of locally compact quantum groups [Kustermans andvan Daele 1997] illustrating nicely some general features of the latter.
However, due to the purely algebraic nature of the theory of multiplier Hopfalgebras it is not possible to treat smooth convolution algebras of general Lie groupsin this context, for instance. The theory of algebraic quantum groups only coversthe case of locally compact groups having compact open subgroups [Landstad andvan Daele 2007]. Accordingly, the variety of quantum groups that can be describedin this setting is obviously limited. It is thus natural to look for a more general setupthen the one provided by algebraic quantum groups.
Motivated by these facts we introduce in this paper the concept of a bornologicalquantum group. The main idea is to replace the category of vector spaces under-lying the definition of an algebraic quantum group by the category of bornologicalvector spaces. It is worth pointing out that bornological vector spaces provide anatural setting for the study of various problems in noncommutative geometry andcyclic homology [Meyer 1999; 2004b; 2006, Voigt 2007].
Whereas the theory of locally convex vector spaces is based on the notion ofan open subset, the key concept in the theory of bornological vector spaces is thenotion of a bounded subset. It is clear from the definitions that both approachesare equivalent for normed spaces. In fact, bornological and topological analysis areessentially equivalent for Frechet spaces [Meyer 2004a]. However, an importantreason to prefer the bornological approach is that the category of bornologicalvector spaces has better algebraic properties than the category of locally convexvector spaces.
For the general theory of bornological quantum groups we follow the work byvan Daele in the algebraic case. Basically, many constructions have to be rephrasedin a more abstract way. This should not be surprising, in fact, the crucial featureof our definition of a bornological quantum group is that it allows us to proveimportant general results while being quite simple at the same time. Althoughstronger assertions are possible at several points of the paper our setup seems tobe sufficiently general for most purposes.
It is natural to ask for the relation between our theory and the operator algebraapproach to quantum groups. Although this will not be discussed in the presentpaper, let us include some remarks on this question. Under suitable additional as-sumptions, including the existence of a ∗-structure, a bornological quantum groupcan be completed to a locally compact quantum group in the sense of Kustermansand Vaes [Kustermans and Vaes 2000]. It is worth pointing out that the locallycompact quantum groups arising in this way are always regular. Thus, from a
BORNOLOGICAL QUANTUM GROUPS 95
slightly different perspective, nonregularity can be viewed as an obstruction tofinding particularly nice subalgebras of “smooth functions” for locally compactquantum groups. This observation might be relevant in the context of the quantumE(2)-group [Woronowicz and Pusz 1999].
We are not aware of an example of a regular locally compact quantum groupthat cannot be obtained by the procedure above. On the other hand, it is clear fromthe examples in Section 10 that different bornological quantum groups may leadto the same locally compact quantum group.
2. Bornological vector spaces
In this section we review basic facts from the theory of bornological vector spaces.More information can be found in [Hogbe-Nlend 1970; 1977; Meyer 1999; 2004a].Throughout we work over the complex numbers.
A bornological vector space is a vector space V together with a collection S(V )of subsets of V satisfying certain conditions. These conditions can be viewed asan abstract reformulation of the properties of bounded subsets in a locally convexvector space. Following [Meyer 1999], we call a subset S of a bornological vectorspace V small if and only if it is contained in the bornology S(V ). Throughout thepaper we only consider bornological vector spaces that are convex and complete.
The guiding example of a bornology is the collection of bounded subsets of alocally convex vector space. We write Bound(V ) for the bornological vector spaceassociated to a locally convex vector space V in this way. One obtains anotherbornological vector space Comp(V ) by considering all precompact subsets of Vinstead. In certain situations the precompact bornology has nicer properties thanthe bounded bornology. Finally, one may view an arbitrary vector space V as abornological vector space by considering the fine bornology Fine(V ) consisting ofthe bounded subsets of finite dimensional subspaces of V .
A linear map f : V → W between bornological vector spaces is called boundedif it maps small subsets to small subsets. The space of bounded linear maps fromV to W is denoted by Hom(V,W ) and carries a natural bornology. In contrast, inthe setting of locally convex vector spaces there are many different topologies onspaces of continuous linear maps.
We point out that the Hahn–Banach theorem does not hold for bornologicalvector spaces. The dual space V ′ of bounded linear functionals on a bornologicalvector space V might very well be zero. A bornological vector space V is calledregular if the bounded linear functionals on V separate points. The regularity ofthe underlying bornological vector space of a bornological quantum group will beguaranteed by the faithfulness of the Haar functional. All examples of bornologicalvector spaces arising from locally convex vector spaces are regular.
96 CHRISTIAN VOIGT
Every complete bornological vector space can be written in a canonical way asa direct limit of Banach spaces. In this way analysis in bornological vector spacesreduces to analysis in Banach spaces.
There exists a natural tensor product in the category of complete bornologicalvector spaces. More precisely, the completed bornological tensor product V ⊗ Wis characterized by the universal property that bounded bilinear maps V ×W → Xcorrespond to bounded linear maps V ⊗ W → X . The bornological tensor productis associative and commutative and there is a natural adjunction isomorphism
Hom(V ⊗ W, X)∼= Hom(V,Hom(W, X))
for all bornological vector spaces V,W, X . This relation is one of the main reasonsthat the category of bornological vector spaces is much better adapted for algebraicconstructions than the category of locally convex spaces. Note that the completedprojective tensor product in the category of locally convex spaces does not have aright adjoint functor because it does not commute with direct sums.
Throughout the paper we will use the leg numbering convention for maps de-fined on tensor products. Moreover, we sometimes write id(n) to indicate that weconsider the identity map on an n-fold tensor product.
A bornological algebra is a complete bornological vector space A with an asso-ciative multiplication given as a bounded linear map µ : A ⊗ A → A. Remark thatbornological algebras are not assumed to have a unit. Modules over bornologicalalgebras and their homomorphisms are defined in the obvious way.
It follows immediately from the definitions that all linear maps f : V → W froma fine space V into any bornological vector space W are bounded. In particularthere is a fully faithful functor Fine from the category of complex vector spacesinto the category of bornological vector spaces. This embedding is compatiblewith tensor products. If V1 and V2 are fine spaces the completed bornologicaltensor product V1⊗V2 is the algebraic tensor product V1 ⊗ V2 equipped with thefine bornology. In particular, every algebra A over the complex numbers can beviewed as a bornological algebra with the fine bornology.
In the case of Frechet spaces a linear map f : V → W is bounded for thebounded or the precompact bornologies if and only if it is continuous. Hence thefunctors Bound and Comp from the category of Frechet spaces into the categoryof bornological vector spaces are fully faithful. The bornological tensor prod-uct of Frechet spaces V,W with the precompact bornology can be identified withthe projective tensor product V ⊗π W equipped with the precompact bornology,see [Meyer 1999].
In our considerations we will need the approximation property in order to avoidcertain analytical problems with completed tensor products. We refer to [Meyer2004a] for detailed information.
BORNOLOGICAL QUANTUM GROUPS 97
Definition 2.1. Let V be a complete bornological vector space. Then V has theapproximation property if the identity map of V can be approximated uniformlyon compact subsets by finite rank operators.
If V is a Frechet space, Grothendieck’s approximation property for V is equiv-alent to the bornological approximation property for Comp(V ) [Meyer 2004a].
We will use the following two properties of bornological vector spaces satisfyingthe approximation property.
Lemma 2.2. Let H be a bornological vector space satisfying the approximationproperty and let ι : V → W be an injective bounded linear map. Then the inducedbounded linear map id ⊗ ι : H ⊗ V → H ⊗ W is injective as well.
Lemma 2.3. Let H be a bornological vector space satisfying the approximationproperty and let V be an arbitrary bornological vector space. Then the canonicallinear map ι : H ⊗ V ′
→ Hom(V, H) is injective.
3. Multiplier algebras
In this section we prove basic results on multiplier algebras of bornological alge-bras that will be needed in the sequel.
The theory of multiplier Hopf algebras is an extension of the theory of Hopfalgebras to the case where the underlying algebras do not have an identity element.Similarly, in our setting we will have to work with nonunital bornological algebras.We will work with bornological algebras that are essential in the following sense.
Definition 3.1. A bornological algebra H is called essential if the multiplicationmap induces an isomorphism H ⊗H H ∼= H .
In order to avoid trivialities we shall always assume that essential bornologicalalgebras are different from zero. Clearly, every unital bornological algebra is es-sential. If H has an approximate identity [Meyer 2004b] then H is essential if andonly if the multiplication H ⊗ H → H is a bornological quotient map. We willnot require the existence of approximate identities in the general definition of abornological quantum group.
Definition 3.2. Let H be a bornological algebra. An H -module V is called essen-tial if the canonical map H ⊗H V → V is an isomorphism.
An analogous definition can be given for right modules. In particular, an essentialalgebra H is an essential left and right module over itself.
We shall now discuss multipliers. A left multiplier for a bornological algebraH is a bounded linear map L : H → H such that L( f g) = L( f )g for all f, g ∈
H . Similarly, a right multiplier is a bounded linear map R : H → H such thatR( f g)= f R(g) for all f, g ∈ H . We let Ml(H) and Mr (H) be the spaces of left
98 CHRISTIAN VOIGT
and right multipliers, respectively. These spaces are equipped with the subspacebornology of Hom(H, H) and become bornological algebras with multiplicationgiven by composition of maps. The multiplier algebra M(H) of a bornologicalalgebra H is the space of all pairs (L , R) where L is a left multiplier and R is aright multiplier for H such that f L(g) = R( f )g for all f, g ∈ H . The bornologyand algebra structure of M(H) are inherited from Ml(H)⊕ Mr (H). There is anatural homomorphism ι : H → M(H). By construction, H is a left and rightM(H)-module.
Let H and K be bornological algebras and let f : H → M(K ) be a homo-morphism. Then K is a left and right H -module in an obvious way. We say thatthe homomorphism f : H → M(K ) is essential if it turns K into an essential leftand right H -module. That is, for the corresponding module structures we haveH ⊗H K ∼= K ∼= K ⊗H H in this case. Note that the identity map id : H → Hdefines an essential homomorphism H → M(H) if and only if the bornologicalalgebra H is essential.
Lemma 3.3. Let H be a bornological algebra and let f : H → M(K ) be anessential homomorphism into the multiplier algebra of an essential bornologicalalgebra K . Then there exists a unique unital homomorphism F : M(H)→ M(K )such that F ι= f where ι : H → M(H) is the canonical map.
Proof. We obtain a bounded linear map Fl : Ml(H)→ Ml(K ) by
Ml(H) ⊗ K ∼= Ml(H) ⊗ H ⊗H Kµ⊗id // H ⊗H K ∼= K
and accordingly a map Fr : Mr (H)→ Mr (K ) by
K ⊗ Mr (H)∼= K ⊗H H ⊗ Mr (H)id⊗µ // K ⊗H H ∼= K .
It is straightforward to check that F((L , R)) = (Fl(L), Fr (R)) defines a unitalhomomorphism F : M(H)→ M(K ) such that F ι = f . Uniqueness of F followsfrom the fact that f (H) · K ⊂ K and K · f (H)⊂ K are dense subspaces. �
The next result, on tensor products, is easily proved.
Lemma 3.4. Let H1, H2 be essential bornological algebras and let f1 : H1 →
M(K1) and f2 : H2 → M(K2) be essential homomorphisms into the multiplieralgebras of bornological algebras K1 and K2. Then the induced homomorphismf1 ⊗ f2 : H1 ⊗ H2 → M(K1 ⊗ K2) is essential.
Following [van Daele 1994], we say that a bornological algebra H is nondegen-erate if f g = 0 for all g ∈ H implies f = 0 and f g = 0 for all f implies g = 0.These conditions are equivalent to saying that the natural maps
H → Ml(H), H → Mr (H)
BORNOLOGICAL QUANTUM GROUPS 99
are injective. In particular, for a nondegenerate bornological algebra the canonicalmap H → M(H) is injective.
Nondegeneracy of a bornological algebra is a consequence of the existence of afaithful linear functional in the following sense.
Definition 3.5. Let H be a bornological algebra. A bounded linear functionalω : H → C is called faithful if ω( f g)= 0 for all g implies f = 0 and ω( f g)= 0for all f implies g = 0.
Remark that a bornological algebra H equipped with a faithful bounded linearfunctional is regular in the sense that bounded linear functionals separate the pointsof H .
Lemma 3.6. Let H1 and H2 be bornological algebras satisfying the approximationproperty equipped with faithful bounded linear functionals φ1 and φ2, respectively.Then φ1 ⊗φ2 is a faithful linear functional on H1 ⊗ H2.
Proof. Assume that x ∈ H1 ⊗ H2 satisfies (φ1 ⊗ φ2)(xy)= 0 for all y ∈ H1 ⊗ H2.Since φ1 is faithful the bounded linear map F1 : H1 → H ′
1 given by F1( f )(g) =
φ1( f g) is injective. Similarly, we have an injective bounded linear map F2 : H2 →
H ′
2 given by F2( f )(g)= φ2( f g). Consider the composition
H1 ⊗ H2// H ′
1 ⊗ H2 // Hom(H1, H2) // Hom(H1, H ′
2),
where the first and third map are induced by F1 and F2, respectively, and thesecond arrow is the obvious one. The first of these maps is injective according toLemma 2.2, the second map is injective according to Lemma 2.3, and injectivityof the third map is obvious. According to our assumption we see that the imageof x in Hom(H1, H ′
2) is zero. Consequently we have x = 0. In a similar way oneshows that (φ1 ⊗φ2)(yx)= 0 for all y ∈ H1 ⊗ H2 implies x = 0. �
4. Bornological quantum groups
In this section we introduce the notion of a bornological quantum group. Moreoverwe prove that every bornological quantum group is equipped with a counit and aninvertible antipode.
In the sequel we assume that H is an essential bornological algebra satisfying theapproximation property. Moreover we suppose that H is equipped with a faithfulbounded linear functional. Remark that we may thus view H as a subset of themultiplier algebra M(H). Taking into account Lemma 3.6, an analogous statementapplies to tensor powers of H .
First we have to discuss the concept of a comultiplication on H . Let 1 : H →
M(H ⊗H) be a homomorphism. The left Galois maps γl, γr : H ⊗H → M(H ⊗H)
100 CHRISTIAN VOIGT
for 1 are defined by
γl( f ⊗ g)=1( f )(g ⊗ 1), γr ( f ⊗ g)=1( f )(1 ⊗ g).
Similarly, the right Galois maps ρl, ρr : H ⊗ H → M(H ⊗ H) for 1 are definedby
ρl( f ⊗ g)= ( f ⊗ 1)1(g), ρr ( f ⊗ g)= (1 ⊗ f )1(g).
The appropriate analogues of these maps play an important role in the algebraicas well as the analytic theory of quantum groups [van Daele 1994; 1998, Baajand Skandalis 1993]. In the operator algebra approach to quantum groups theycorrespond to multiplicative unitaries [Baaj and Skandalis 1993]. Our terminologyis motivated from the fact that variants of these maps also occur in the theory ofHopf–Galois extensions; see for instance [Montgomery 1993].
Assume in addition that the homomorphism 1 : H → M(H ⊗ H) is essential.Then 1 is called coassociative if
(1 ⊗ id)1= (id ⊗1)1,
where both sides are viewed as maps from H to M(H ⊗ H ⊗ H). These maps arewell-defined according to Lemma 3.4.
Definition 4.1. An essential homomorphism 1 : H → M(H ⊗ H) is called acomultiplication if it is coassociative.
An essential algebra homomorphism f : H → M(K ) between bornologicalalgebras equipped with comultiplications is called a coalgebra homomorphism if1 f = ( f ⊗ f )1.
We need some more terminology. The opposite algebra H op of H is the spaceH equipped with the opposite multiplication. That is, the multiplication µop inH op is defined by µop
= µτ where µ : H ⊗ H → H is the multiplication in Hand τ : H ⊗ H → H ⊗ H is the flip map given by τ( f ⊗ g)= g ⊗ f . An algebraantihomomorphism between H and K is an algebra homomorphism φ : H →
K op. Equivalently, an algebra antihomomorphism can be viewed as an algebrahomomorphism H op
→ K . If 1 : H → M(H ⊗ H) is a comultiplication then 1also defines a comultiplication H op
→ M(H op⊗H op). We write γ op
l , γopr , ρ
opl , ρ
opr
for the corresponding Galois maps.Apart from changing the order of multiplication we may also reverse the order of
a comultiplication. If 1 : H → M(H ⊗ H) is a comultiplication then the oppositecomultiplication1cop is the essential homomorphism from H to M(H⊗H) definedby 1cop
= τ1. We write γ copl , γ
copr , ρ
copl , ρ
copr for the Galois maps associated to
this comultiplication. Moreover we write H cop for H equipped with the oppo-site comultiplication. Using opposite comultiplications we obtain the notion of acoalgebra antihomomorphism.
BORNOLOGICAL QUANTUM GROUPS 101
We may also combine these procedures, that is, reverse both multiplication andcomultiplication. The bornological algebra with comultiplication arising in thisway is denoted by H op cop
= (H op)cop and we write γ op copl , γ
op copr , ρ
op copl , ρ
op copr
for the corresponding Galois maps.It is straightforward to check that the Galois maps of H, H op, H cop and H op cop
are related as follows.
Lemma 4.2. Let 1 : H → M(H ⊗ H) be a comultiplication. Then
γr = τγcopl , ρl = γ
opl τ, ρr = τγ
op copl τ,
γl = τγ copr , ρr = γ op
r τ, ρl = τγ op copr τ,
ρr = τρcopl , γl = ρ
opl τ, γr = τρ
op copl τ,
ρl = τρcopr , γr = ρop
r τ, γl = τρop copr τ
for the Galois maps of H, H op, H cop and H op cop.
These relations can also be rewritten in the form
γcopl = τγr , γ
opl = ρlτ, γ
op copl = τρrτ,
γ copr = τγl, γ op
r = ρrτ, γ op copr = τρlτ,
ρcopl = τρr , ρ
opl = γlτ, ρ
op copl = τγrτ,
ρcopr = τρl, ρop
r = γrτ, ρop copr = τγlτ.
As a consequence, the Galois maps for H may be expressed in terms of the mapsγl, γ
opl , γ
copl and γ op cop
l and vice versa. Of course, there are similar statements forγr , ρl and ρr . This basic observation will be used frequently below.
Let 1 : H → M(H ⊗ H) be a comultiplication such that all Galois maps asso-ciated to 1 define bounded linear maps from H ⊗ H into itself. If ω is a boundedlinear functional on H we define for every f ∈ H a multiplier (id⊗ω)1( f )∈ M(H)by
(id ⊗ω)1( f ) · g = (id ⊗ω)γl( f ⊗ g)
g · (id ⊗ω)1( f )= (id ⊗ω)ρl(g ⊗ f ).
To check that this is indeed a two-sided multiplier observe that
( f ⊗ 1)γl(g ⊗ h)= ρl( f ⊗ g)(h ⊗ 1)
for all f, g, h ∈ H . In a similar way we define (ω ⊗ id)1( f ) ∈ M(H) by
(ω ⊗ id)1( f ) · g = (id ⊗ω)γr ( f ⊗ g)
g · (ω ⊗ id)1( f )= (id ⊗ω)ρr (g ⊗ f ).
102 CHRISTIAN VOIGT
Definition 4.3. Let1 : H → M(H ⊗ H) be a comultiplication such that all Galoismaps associated to 1 define bounded linear maps from H ⊗ H into itself.
A bounded linear functional φ : H → C is called left invariant if
(id ⊗φ)1( f )= φ( f )1
for all f ∈ H . Similarly, a bounded linear functional ψ : H → C is called rightinvariant if
(ψ ⊗ id)1( f )= ψ( f )1
for all f ∈ H .
We now give the definition of a bornological quantum group.
Definition 4.4. A bornological quantum group is an essential bornological algebraH satisfying the approximation property together with a comultiplication1 : H →
M(H ⊗ H) such that all Galois maps associated to 1 are isomorphisms and afaithful left invariant functional φ : H → C.
A morphism between bornological quantum groups H and K is an essentialalgebra homomorphism α : H → M(K ) such that (α ⊗α)1=1α.
To be precise, the Galois maps in a bornological quantum group are supposed toyield bornological isomorphisms of H ⊗ H into itself. The left invariant functionalφ is also referred to as the left Haar functional.
Our definition of a bornological quantum group is equivalent to the definition ofan algebraic quantum group in the sense of van Daele [van Daele 1998] providedthe underlying bornological vector space carries the fine bornology. The only dif-ference in this case is that we have included faithfulness of the Haar functional inthe definition.
An easy computation yields the following assertion.
Lemma 4.5. Let H be a bornological quantum group. Then
(ρl ⊗ id)(id ⊗ γr )= (id ⊗ γr )(ρl ⊗ id)
where both sides are viewed as maps from H ⊗ H ⊗ H into itself.
The next theorem provides an alternative description of bornological quantumgroups.
Theorem 4.6. Let H be an essential bornological algebra satisfying the approx-imation property and let 1 : H → M(H ⊗ H) be a comultiplication such thatall associated Galois maps define bounded linear maps from H ⊗ H to itself.Moreover assume that φ : H → C is a faithful left invariant functional. ThenH is a bornological quantum group if and only if there exist an essential algebra
BORNOLOGICAL QUANTUM GROUPS 103
homomorphism ε : H → C and a linear isomorphism S : H → H which is both analgebra antihomomorphism and a coalgebra antihomomorphism such that
(ε ⊗ id)1= id = (id ⊗ ε)1
andµ(S ⊗ id)γr = ε ⊗ id, µ(id ⊗ S)ρl = id ⊗ ε.
In this case the maps ε and S are uniquely determined.
Proof. The proof follows the discussion in [van Daele 1994]. Along the way weobtain some formulas which are also useful in other situations.
Let us first assume that there exist maps ε and S satisfying the above conditions.Following the traditional terminology, these maps will be called the counit and theantipode of H . We claim that the inverse γ−1
r of γr is given by
γ−1r = (S−1
⊗ id)γ copr (S ⊗ id).
Using that S is a coalgebra antihomomorphism we obtain the equality
(S−1⊗ id)γ cop
r (S ⊗ id)= (S−1⊗ id)(id ⊗µ)(1cop
⊗ id)(S ⊗ id)
= (id ⊗µ)(S−1⊗ id ⊗ id)(τ1S ⊗ id)
= (id ⊗µ)(id ⊗ S ⊗ id)(1 ⊗ id),
where both sides are viewed as maps from H ⊗ H to M(H ⊗ H). In particular,the image of the last map is contained in H ⊗ H . We compute
µ(2)(γ−1r γr ⊗ id(2))= µ(2)(id ⊗µ ⊗ id(2))(id ⊗ S ⊗ id(3))(1 ⊗ id(3))(γr ⊗ id(2))
= µ(2)(id ⊗ ε ⊗ id(3))(1 ⊗ id(3))= µ(2),
where we write µ(2) for the multiplication in the tensor product H ⊗H . In a similarway one shows µ(2)(id(2) ⊗γrγ
−1r )=µ(2) which shows that γr is an isomorphism.
For the other Galois maps one could perform similar calculations. We proceeddifferently and show first that the given counit and invertible antipode for H yieldcounits and antipodes for H op, H cop and H op cop as well. Observe that the counitε satisfies
(ε ⊗ id)1cop= id = (id ⊗ ε)1cop,
which means that ε is a counit for H cop and H op cop. Using Lemma 4.2 we obtain
µop(S ⊗ id)γ op copr = µτ(S ⊗ id)τρlτ = µ(id ⊗ S)ρlτ = ε ⊗ id,
µop(id ⊗ S)ρop copl = µτ(id ⊗ S)τγrτ = µ(id ⊗ S)γrτ = id ⊗ ε,
which shows that S is an antipode for H op cop. A computation shows that
µ(µop⊗ id)(S−1
⊗ id ⊗ id)(γ opr ⊗ id)= µ(ε ⊗ id ⊗ id),
104 CHRISTIAN VOIGT
which yieldsµop(S−1
⊗ id)γ opr = ε ⊗ id.
Similarly we have µ(id ⊗µop)(id ⊗ id ⊗ S−1)(id ⊗ ρopl ) = µ(id ⊗ id ⊗ ε), which
impliesµop(id ⊗ S−1)ρ
opl = id ⊗ ε.
Hence S−1 is an antipode for H op. As above it follows that S−1 is also an antipodefor H cop
= (H op)op cop. We may now apply our previous argument for the Galoismap γr to H op, H cop and H op cop and use Lemma 4.2 to see that γl, ρl and ρr areisomorphisms as well. This shows that H is a bornological quantum group.
Conversely, let us assume that H is a bornological quantum group and constructthe maps ε and S. We begin with the counit ε. Choose an element h ∈ H such thatφ(h)= 1 and set
ε( f )= φ(µρ−1l (h ⊗ f )).
This obviously yields a bounded linear map ε : H → C. Using
(4-1) γr (id ⊗µ)= (id ⊗µ)γr ,
we easily see that the formula E( f ) · g = µγ−1r ( f ⊗ g) defines a left multiplier
E( f ) of H . Actually, we obtain a bounded linear map E : H → Ml(H) in thisway. Using Lemma 4.5 we obtain
(id ⊗µ)(id ⊗ E ⊗ id)(ρl ⊗ id)(id ⊗ γr )= (id ⊗µ)(id ⊗ γ−1r )(ρl ⊗ id)(id ⊗ γr )
= (id ⊗µ)(id ⊗ γ−1r )(id ⊗ γr )(ρl ⊗ id)
= (µ ⊗ id)(id ⊗ γr ).
Since γr and ρl are isomorphisms, this implies
(4-2) (id ⊗µ)(id ⊗ E ⊗ id)= µρ−1l ⊗ id.
Evaluating (4-2) on a tensor h ⊗ f ⊗ g where h is chosen as above and applyingφ ⊗ id we get
E( f ) · g = (φ ⊗ id)(h ⊗ E( f ) · g)= φ(µρ−1l (h ⊗ f ))g = ε( f )g
and hence
(4-3) E( f )= ε( f )1
in Ml(H) for every f ∈ H . Thus we could have used any nonzero bounded linearfunctional in order to define ε.
According to (4-3) and the definition of E we have
(4-4) (ε ⊗ id)γr = µ(E ⊗ id)γr = µ.
BORNOLOGICAL QUANTUM GROUPS 105
Equation (4-2) yields gε( f )⊗h = g ⊗ε( f )h =µρ−1l (g ⊗ f )⊗h for all f, g ∈ H ,
which implies
(4-5) gε( f )= µρ−1l (g ⊗ f ).
This is equivalent to
(4-6) (id ⊗ ε)ρl = µ,
since ρl is an isomorphism.We now show that ε is an algebra homomorphism. We have
(4-7) ρl(id ⊗µ)= (µ ⊗ id)(id ⊗µ(2))(id ⊗1 ⊗1)= µ(2)(ρl ⊗1),
because 1 is an algebra homomorphism. From this relation and (4-6) we get
(id ⊗ ε)µ(2)(id ⊗ id ⊗1)(ρl ⊗ id)= (id ⊗ ε)ρl(id ⊗µ)= µ(id ⊗µ)
= µ(µ ⊗ id)= µ(id ⊗ ε ⊗ id)(ρl ⊗ id),
and since ρl is an isomorphism this implies
(id ⊗ ε)µ(2)(id ⊗ id ⊗1)= µ(id ⊗ ε ⊗ id).
Now observe µ(2)(id ⊗ id ⊗1)= (id ⊗µ)ρ13l and hence
(id ⊗ ε)(id ⊗µ)ρ13l = µ(id ⊗ ε ⊗ id)= (id ⊗ ε)(id ⊗ ε ⊗ id)ρ13
l ,
where we use (4-6). We deduce (id ⊗ ε)(id ⊗µ) = (id ⊗ ε)(id ⊗ ε ⊗ id), whichimplies ε( f g)= ε( f )ε(g) for all f, g ∈ H . Thus ε is an algebra homomorphism.
Using this fact and (4-6) we calculate
µ(id(2)⊗ε)(id⊗γr )= (id⊗ε)(id⊗µ)(ρl ⊗id)= (id⊗µ)(id⊗ε⊗ε)(ρl ⊗id)=µ⊗ε,
which implies
(4-8) (id ⊗ ε)γr = id ⊗ ε.
Analogously,
(4-9) (ε ⊗ id)ρl = ε ⊗ id,
as a consequence of (4-4).It is easy to see that the map ε is nonzero. To check that ε is nondegenerate we
define a bounded linear map σ : C → H ⊗H C by σ(1)= k ⊗1 where k ∈ H is anelement satisfying ε(k)= 1. Using (4-4) and (4-8) we obtain
σ(ε⊗ id)( f ⊗ 1)= ε( f )k ⊗ 1 = µγ−1r ( f ⊗ k)⊗ 1
= (id ⊗ ε)γ−1r ( f ⊗ k)= f ⊗ ε(k)1 = f ⊗ 1,
106 CHRISTIAN VOIGT
which implies H ⊗H C ∼= C. In a similar way one checks C⊗H H ∼= C using (4-9).This shows that ε is nondegenerate.
According to (4-6) we thus have
(4-10) (id ⊗ ε)1= id,
and using (4-4) we get
(4-11) (ε ⊗ id)1= id.
Conversely, the last equation implies (ε ⊗ id)γr = µ which in turn determines εuniquely since γr is an isomorphism.
Now we shall construct the antipode. It is easy to check that the formulas
Sl( f ) · g = (ε ⊗ id)γ−1r ( f ⊗ g), g · Sr ( f )= (id ⊗ ε)ρ−1
l (g ⊗ f )
define a left multiplier Sl( f ) and a right multiplier Sr ( f ) of H for every f ∈ H . Inthis way we obtain bounded linear maps Sl : H → Ml(H) and Sr : H → Mr (H).
Let us show that Sl is an algebra antihomomorphism. Using Lemma 4.5 and(4-6) we calculate
(id ⊗µ)(id ⊗ Sl ⊗ id)(ρl ⊗ id)= (µ ⊗ id)(id ⊗ γ−1r ).
Applying the multiplication map µ to this equation yields
(4-12) µ(id ⊗µ)(id ⊗ Sl ⊗ id)(ρl ⊗ id)= µ(id ⊗µγ−1r )= µ(id ⊗ ε ⊗ id),
where we have used (4-4). Applying (4-7), (4-12), the fact that ε is an algebrahomomorphism, and again (4-12), we obtain
µ(id⊗µ)(id ⊗ Sl ⊗ id)(µ(2) ⊗ id)(ρl ⊗1 ⊗ id)
= µ(id ⊗µ)(id ⊗ Sl ⊗ id)(ρl ⊗ id)(id ⊗ id ⊗ ε ⊗ id).
Since ρl is an isomorphism this yields, due to (4-12),
µ(id ⊗µ)(id ⊗ Sl ⊗ id)(id ⊗µ ⊗ id)ρ13l
= µ(id ⊗µ)(id ⊗ Sl ⊗ id)(id(2) ⊗µ)(id(2) ⊗ Sl ⊗ id)(id ⊗ τ ⊗ id)ρ13l ,
and hence
µ(id ⊗µ)(id ⊗ Sl ⊗ id)(id ⊗µ ⊗ id)
= µ(id ⊗µ)(id ⊗µ ⊗ id)(id ⊗ τ ⊗ id)(id ⊗ Sl ⊗ Sl ⊗ id).
Since the algebra H is nondegenerate we obtain
(4-13) Sl( f g)= Sl(g)Sl( f )
for all f, g ∈ H , as claimed.
BORNOLOGICAL QUANTUM GROUPS 107
For the map Sr we do an analogous calculation. We have
(µ ⊗ id)(id ⊗ Sr ⊗ id)(id ⊗γr )= (µ ⊗ id)(id ⊗ Sr ⊗ id)(ρl ⊗ id)(id ⊗γr )(ρ−1l ⊗ id)
= (id ⊗ε ⊗ id)(id ⊗γr )(ρ−1l ⊗ id)
= (id ⊗µ)(ρ−1l ⊗ id),
and applying µ yields
(4-14) µ(µ ⊗ id)(id ⊗ Sr ⊗ id)(id ⊗ γr )= µ(µρ−1l ⊗ id)= µ(id ⊗ ε ⊗ id).
As above one may proceed to show that Sr is an algebra antihomomorphism. Weshall instead first show that (Sl( f ), Sr ( f )) is a two-sided multiplier of H for everyf ∈ H . By the definition of Sr we have µ(id⊗ Sr )= (id⊗ε)ρ−1
l and hence (4-12)implies
µ(id ⊗µ)(id ⊗ Sl ⊗ id)= µ(id ⊗ ε ⊗ id)(ρ−1l ⊗ id)= µ(µ ⊗ id)(id ⊗ Sr ⊗ id),
which is precisely the required identity. We can now use (4-13) to obtain thatSr is an algebra antihomomorphism. If S : H → M(H) denotes the linear mapdetermined by Sl and Sr we have thus showed so far that S : H → M(H) is abounded algebra antihomomorphism.
For f ∈ H , define Sl( f ) ∈ Ml(H) and Sr ( f ) ∈ Mr (H) by
Sl( f ) · g = (ε ⊗ id)γ−1l τ( f ⊗ g),
g · Sr ( f )= (id ⊗ ε)ρ−1r τ(g ⊗ f ).
According to Lemma 4.2 we have γ−1l τ = (γ
copr )−1 and ρ−1
r τ = (ρcopl )−1. The
discussion above applied to H cop shows that Sl and Sr determine a bounded algebraantihomomorphism S : H → M(H).
Our next goal is to prove that S and S actually define bounded linear maps fromH into itself which are inverse to each other. To do this, observe that
(id ⊗µ)(τ ⊗ id)= (id ⊗µ)(τ ⊗ id)(γr ⊗ id)(γ−1r ⊗ id)
= µ(2)(1cop
⊗ id(2))(γ−1r ⊗ id),
which implies
(µ ⊗ id)(id ⊗ S ⊗ id)(id ⊗ id ⊗µ)(id ⊗ τ ⊗ id)
= (µ ⊗ id)(id ⊗µ ⊗ id)(id ⊗ S ⊗ S ⊗ id)(id(2) ⊗ γ copr ⊗ id)(id ⊗ τγ−1
r ⊗ id),
since S is an algebra antihomomorphism. Applyingµ to this equation and insertingthe definitions of S and S we calculate
µ(id ⊗µ)(id ⊗µ ⊗ id)(id ⊗ S ⊗ id(2))(id ⊗ τ ⊗ id)
= µ(id ⊗µ)(id ⊗ S ⊗ id)(id ⊗µ ⊗ id)(id ⊗ S ⊗ id(2)).
108 CHRISTIAN VOIGT
As a consequence we obtain the relation
(4-15) µ(S ⊗ id)τ = Sµ(S ⊗ id),
where both sides are viewed as maps from H ⊗ H into M(H). Choose k ∈ H suchthat ε(k)= 1. Equation (4-15) together with the definition of S yields the relation
S( f )= S( f )ε(k)= Sµ(S ⊗ id)γr (k ⊗ f )= µ(S ⊗ id)τγr (k ⊗ f )
for all f ∈ H . This shows that S defines a bounded linear map from H to H .Replacing H by H cop we see that S may be viewed as a bounded linear map fromH to H as well. Since S is an algebra antihomomorphism, (4-15) then yieldsµ(S ⊗ id)= µ(S ⊗ SS) and hence
µ(µ ⊗ id)(id ⊗ S ⊗ id)= µ(µ ⊗ id)(id ⊗ S ⊗ SS).
Since µ(id⊗ S)= (id⊗ε)ρ−1r τ by the definition of S we get SS = id. Analogously
one obtains SS = id. Equations (4-14) and (4-12) yield
µ(S ⊗ id)γr = ε ⊗ id, µ(id ⊗ S)ρl = id ⊗ ε,
as desired. Moreover these equations determine the map S uniquely.We show that S is a coalgebra antihomomorphism. Since 1 is an algebra ho-
momorphism we have
γr (µ ⊗ id)= (id ⊗µ)(1 ⊗ id)(µ ⊗ id)= (id ⊗µ)(γl ⊗ id)(id ⊗ γr )
and using (4-1) we get
(4-16) (µ ⊗ id)(id ⊗ γ−1r )= (id ⊗µ)(γ−1
r ⊗ id)(γl ⊗ id).
According to Lemma 4.5, (4-6) and the definition of S we get
(µ ⊗ id)(id ⊗ γ−1r )= (id ⊗µ)(id ⊗ S ⊗ id)(ρl ⊗ id).
Together with (4-16) we obtain
(4-17) γ−1r γl = (id ⊗ S)ρl,
which, applied to H cop, yields
(4-18) γ−1l γr = (γ cop
r )−1ττγcopl = (id ⊗ S−1)ρ
copl = (id ⊗ S−1)τρr .
Equations (4-17) and (4-18) imply
(4-19) ρr (S ⊗ S)= (S ⊗ id)τρ−1l (S ⊗ id).
From the definition of S and (4-4) we have
(µ ⊗ id)(id ⊗ S ⊗ id)(id ⊗ γr )(ρl ⊗ id)= (id ⊗ ε ⊗ id)(id ⊗ γr )= (id ⊗µ).
BORNOLOGICAL QUANTUM GROUPS 109
This implies
(id ⊗µ)(ρ−1l ⊗ id)(τ ⊗ id)(id ⊗ S ⊗ id)= (id ⊗µ)(S ⊗ id(2))(γl ⊗ id),
since S is an algebra antihomomorphism. Consequently we obtain
(4-20) (S ⊗ id)γl = ρ−1l (S ⊗ id)τ.
Equations (4-19) and (4-20) yield
ρr (S ⊗ S)τ = (S ⊗ id)τρ−1l (S ⊗ id)τ = (S ⊗ S)τγl,
and using that S is an algebra antihomomorphism another computation gives
(id ⊗µop)(1 ⊗ id)(S ⊗ id)(id ⊗ S)
= (id ⊗µop)(S ⊗ S ⊗ id)(τ ⊗ id)(1 ⊗ id)(id ⊗ S).
This implies1S = (S ⊗S)τ1 and shows that S is a coalgebra antihomomorphism.Of course S−1 is a coalgebra antihomomorphism as well. Thus we have shownthat there exist unique maps S and ε with the desired properties. �
Recall that a morphism of bornological quantum groups is an essential algebrahomomorphism α : H → M(K ) which is also a coalgebra homomorphism.
Proposition 4.7. Every morphism α : H → M(K ) of bornological quantum groupsis automatically compatible with the counits and the antipodes.
Proof. Note that the Galois maps associated to the comultiplication of H extendto bounded linear maps from M(H) ⊗ M(H) into M(H ⊗ H). Moreover observethat the relation (ε ⊗ id)γr =µ obtained in (4-4) still holds when we consider bothsides as maps form M(H) ⊗ M(H) to M(H).
Since α : H → M(K ) is an algebra homomorphism and a coalgebra homomor-phism we have
(4-21) γr (α ⊗α)= (α ⊗α)γr ,
where both sides are viewed as maps from H ⊗ H into M(K ⊗ K ). Hence weobtain
(ε ⊗ id)(α ⊗α)γr = (ε ⊗ id)γr (α ⊗α)= µ(α ⊗α)= αµ= (ε ⊗α)γr ,
where all maps are considered to be defined on H ⊗ H with values in M(K ). Weconclude (εα)⊗α= ε ⊗α because γr is an isomorphism. Since α is nondegeneratethis shows εα = ε which means that α is compatible with the counits.
The arguments given in the proof of Theorem 4.6 show that the inverses of theGalois maps of H can be described explicitly using the antipode S and its inverse.It follows that these maps are defined on M(H) ⊗ M(H) in a natural way.
110 CHRISTIAN VOIGT
With this in mind and using the equality
(µ ⊗ id)(id ⊗ ρr )(id ⊗µ ⊗ id)(id ⊗ τ ⊗ id)(ρl ⊗ id(2))
= µ(2)(id(2) ⊗µ(2))(id(2) ⊗1 ⊗1)(id ⊗ τ ⊗ id),
we compute on H ⊗ H ⊗ M(H) ⊗ H the relation
(µ ⊗ id)(id ⊗µ ⊗ id)(id(2) ⊗ γ−1r )= (µ ⊗ id)(id ⊗ γ−1
r )(id ⊗µ ⊗ id)γ 24r .
This yields (µ ⊗ id)(id ⊗ γ−1r ) = γ−1
r (µ ⊗ id)γ 13r on H ⊗ M(H) ⊗ H , which in
turn implies
(4-22) (µ ⊗µ)(id ⊗ γ−1r ⊗ id)= γ−1
r (µ ⊗ id)γ 13r (id(2) ⊗µ)
if both sides are viewed as maps from H ⊗ M(H) ⊗ M(H) ⊗ H into H ⊗ H .Moreover,
γr (µ ⊗µ)(id ⊗α ⊗α ⊗ id)
= (id ⊗µ)(id(2) ⊗µ)(1 ⊗ id(2))(µ ⊗ id(2))(id ⊗α ⊗α ⊗ id)
= (µ ⊗ id)γ 13r (id(2) ⊗µ)(id ⊗α ⊗α ⊗ id)(id ⊗ γr ⊗ id)
as maps from K ⊗ H ⊗ H ⊗ K into K ⊗ K , which yields, according to (4-22),
(µ ⊗µ)(id ⊗α ⊗α ⊗ id)= γ−1r γr (µ ⊗µ)(id ⊗α ⊗α ⊗ id)
= (µ ⊗µ)(id ⊗γ−1r ⊗ id)(id ⊗α ⊗α ⊗ id)(id ⊗γr ⊗ id)
and we deduce
(4-23) γ−1r (α ⊗α)= (α ⊗α)γ−1
r .
Note that this assertion does not immediately follow from (4-21) since the mapγ−1
r is not defined on the multiplier algebra M(K ⊗ K ).Now, the relation µ(S ⊗ id) = (ε ⊗ id)γ−1
r still holds if both sides are viewedas maps from M(H) ⊗ M(H) into M(H). Using this observation we obtain therelations
µ(S ⊗ id)(α ⊗α)= (ε ⊗ id)γ−1r (α ⊗α),
αµ(S ⊗ id)= (ε ⊗α)γ−1r ,
which, together with (4-23) and the fact that α is compatible with the counits, yield
µ(αS ⊗α)= αµ(S ⊗ id)= (ε ⊗ id)(α ⊗α)γ−1r = (ε ⊗ id)γ−1
r (α ⊗α)=µ(Sα ⊗α).
Since α is nondegenerate it follows that the maps αS and Sα coincide. Remarkthat we also have αS−1
= S−1α. �
BORNOLOGICAL QUANTUM GROUPS 111
Proposition 4.7 implies in particular εS = ε and εS−1= ε in the bornologi-
cal quantum group H since S : H → H op cop is an isomorphism of bornologicalquantum groups.
5. Modular properties of the integral
In this section we discuss modular properties of the Haar functional on a bornolog-ical quantum group. Since the results and arguments are parallel to the ones foralgebraic quantum groups [van Daele 1994; 1998] we omit the proofs.
Let H be a bornological quantum group. If φ is a left invariant functional thenS(φ) is right invariant where S(φ)( f )=φ(S( f )). In particular, there always existsa faithful right invariant functional on H .
Proposition 5.1. Let H be a bornological quantum group and let φ and ψ befaithful left and right invariant functionals on H, respectively. There exists abornological isomorphism ν of H such that
ψ(h f )= φ(hν( f ))
for all f, h ∈ H.
Proposition 5.2. The left Haar functional φ for a bornological quantum group His unique up to a scalar.
Proposition 5.3. Let H be a bornological quantum group. There exists a uniquebounded algebra automorphism σ of H such that
φ( f g)= φ(gσ( f ))
for all f, g ∈ H. Moreover φ is invariant under σ .
Let ψ be a right Haar measure on H . Then S(ψ) is a left Haar measure and weobtain
ψ(S( f )S(g))= S(ψ)(g f )= S(ψ)(σ−1( f )g)= ψ(S(g)(Sσ−1S−1)S( f )))
for all f, g ∈ H , and thusψ( f g)= ψ(gρ( f )),
where ρ = Sσ−1S−1. If we also consider S−1(ψ) and use Proposition 5.2 we getρ = S−1σ−1S which yields the relation S2σ = σ S2 for the automorphism σ .
Proposition 5.4. Let H be a bornological quantum group. There exists a uniquemultiplier δ ∈ M(H) such that
(φ ⊗ id)1( f )= φ( f )δ
for all f ∈ H.
112 CHRISTIAN VOIGT
The multiplier δ is called the modular element of H .
Proposition 5.5. The modular element δ is invertible and satisfies the relations
1(δ)= δ⊗ δ, ε(δ)= 1, S(δ)= δ−1.
Observe that we also have S−1(δ)= δ−1. If ψ is a faithful right invariant func-tional then the formula
(id ⊗ψ)1( f )= ψ( f )δ−1
describes the corresponding modular relation. This follows from Propositions 5.4and 5.5 using the left invariant functional φ = S(ψ).
6. Modules and comodules
In this section we discuss the concepts of an essential module and an essentialcomodule over a bornological quantum group.
We begin with the notion of an essential module. Actually, the definition ofessential modules over bornological algebras was already given in Section 3.
Definition 6.1. Let H be a bornological quantum group. An essential H -moduleis an H -module V such that the module action induces a bornological isomorphismH ⊗H V ∼= V . A bounded linear map f : V → W between essential H -modules iscalled H -linear if µW (id ⊗ f )= f µV .
If H and the H -module V carry the fine bornology, then V is essential if andonly if H V = V . This follows easily from the fact that H has an approximateidentity in this case. Modules satisfying the condition H V = V are called unitalin [Drabant et al. 1999]. Hence unital modules over an algebraic quantum groupare essential.
We denote the category of essential H -modules by H-Mod. By definition, themorphisms in H-Mod are the bounded H -linear maps.
The direct sum of a family of essential H -modules is again an essential H -module. Moreover, the tensor product V ⊗W of two essential H -modules becomesan essential H -module using the diagonal action. The trivial one-dimensional H -module C given by the counit ε behaves like a unit with respect to the tensorproduct.
Dually to the concept of an essential module one has the notion of an essentialcomodule. Let H be a bornological quantum group, let V be a bornological vectorspace and let HomH (H, V ⊗H) be the space of bounded right H -linear maps fromH to V ⊗H . A coaction of H on V is a bounded linear map η : V →HomH (H, V ⊗
H)which is colinear in the following sense. By adjoint associativity, the map η canequivalently be described as a bounded H -linear map V ⊗ H → V ⊗ H . Then η is
BORNOLOGICAL QUANTUM GROUPS 113
said to be H -colinear if the latter map is an isomorphism and satisfies the relation
(id ⊗ γr )η12(id ⊗ γ−1r )= η12η13
where both sides are viewed as maps from V ⊗ H ⊗ H to itself.
Definition 6.2. Let H be a bornological quantum group. An essential H -comoduleis a bornological vector space V together with a coaction η : V → HomH (H, V ⊗
H). A bounded linear map f : V → W between essential comodules is calledH -colinear if ( f ⊗ id)ηV = ηW ( f ⊗ id).
We write Comod-H for the category of essential comodules over H with H -colinear maps as morphisms. More precisely, we have defined right comodules.There are analogous definitions for left comodules. Let us point out that corepre-sentations and comodules in the framework of multiplier Hopf algebras have beendiscussed in detail in [van Daele and Zhang 1999; Kurose et al. 2000].
The most elementary example of a coaction is the trivial coaction τ of H on V .The map τ : V → HomH (H, V ⊗ H) is given by τ(v)( f )= v⊗ f . Equivalently,the linear map V ⊗ H → V ⊗ H corresponding to τ is the identity.
As in the case of essential modules, there exists a tensor product in the categoryof essential comodules. Assume that ηV :V ⊗H →V ⊗H and ηW :W ⊗H →W ⊗Hare essential comodules. Then the tensor product coaction ηV ⊗W is defined as thecomposition
V ⊗ W ⊗ Hη23
W // V ⊗ W ⊗ Hη13
V // V ⊗ W ⊗ H.
It is clear that ηV ⊗W is a right H -linear isomorphism and a straightforward calcu-lation shows that it is indeed a coaction. The trivial coaction on C behaves like aunit with respect to the tensor product of comodules.
An important example of a coaction is the regular coaction of H on itself givenby the comultiplication 1 : H → M(H ⊗ H). More precisely, the regular coactionis the map from H to HomH (H, H ⊗ H) corresponding to the Galois map γr . Therelation
(id ⊗ γr )γ12r (id ⊗ γ−1
r )= γ 12r γ 13
r
is easily verified. Note that this corresponds to the pentagon equation of the Kac–Takesaki operator [Baaj and Skandalis 1993].
Consider the special case that the bornological quantum group H is unital. Thenthere is a natural isomorphism HomH (H, V ⊗ H) ∼= V ⊗ H and a coaction is thesame thing as a bounded linear map η : V → V ⊗ H such that (η⊗id)η= (id⊗1)η
and (id ⊗ ε)η = id. That is, for unital bornological quantum groups the notion ofa coaction is very similar to the concept of a coaction as it is used in the theory ofHopf algebras.
114 CHRISTIAN VOIGT
For later use we give the following definitions. An essential H -module P overa bornological quantum group H is called projective if for every H -linear mapπ : V → W with bounded linear splitting σ : W → V and every H -linear mapξ : P → W there exists an H -linear map ζ : P → V such that πζ = ξ . In thiscase we say that P satisfies the lifting property for linearly split surjections of H -modules. In a completely analogous way one defines the notion of a projectiveessential H -comodule.
We conclude this section by studying the functoriality of essential modules andcomodules under morphisms of quantum groups. Let α : H → M(K ) be a mor-phism of bornological quantum groups. If λ : K ⊗V → V is an essential K -modulestructure on V then α∗(λ) is the H -module structure defined by
H ⊗ Vα⊗id // M(K ) ⊗ V
∼= // M(K ) ⊗ K ⊗K Vµ⊗id // K ⊗K V ∼= V,
and it is easy to check that V becomes an essential H -module in this way. Thisconstruction is evidently compatible with module maps and thus yields a functorα∗
: K-Mod → H-Mod. A similar functor is obtained for right modules.Conversely, let η : V ⊗ H → V ⊗ H be an essential H -comodule. We define a
bounded linear map α∗(η) : V ⊗ K → V ⊗ K by the commutative diagram
V ⊗ Kα∗(η) //
∼=
��
V ⊗ K
∼=
��V ⊗ H ⊗H K
η⊗id // V ⊗ H ⊗H K
where we use that η is right H -linear. It is evident that α∗(η) is a right K -linearisomorphism and one checks that the relation
(id ⊗ γr )α∗(η)12(id ⊗ γ−1r )= α∗(η)12α∗(η)13
is satisfied. Hence α∗(η) defines a coaction of K on V . This construction is com-patible with comodule maps and yields a functor α∗ : Comod-H → Comod-K .Again, there is a similar functor for left comodules.
7. The dual quantum group and Pontrjagin duality
In this section we construct the dual quantum group H of a bornological quantumgroup H . Moreover we prove the analogue of Pontrjagin duality in the context ofbornological quantum groups. Unless further specified we assume that φ is a leftHaar functional on H and we let ψ be any right Haar functional.
BORNOLOGICAL QUANTUM GROUPS 115
Using the invariant functional φ we define bounded linear maps Fl and Fr fromH into the dual space H ′
= Hom(H,C) by
Fl( f )(h)= φ(h f ), Fr ( f )(h)= φ( f h).
Similarly, we obtain bounded linear maps Gl and Gr from H into H ′ by
Gl( f )(h)= ψ(h f ), Gr ( f )(h)= ψ( f h),
and all these maps are injective by faithfulness. Using notation and results fromSection 5 we obtain:
Proposition 7.1. Let H be a bornological quantum group. Then
Fr ( f )= Fl(σ ( f )), Gl( f )= Fl(ν( f )), Gr ( f )= Gl(ρ( f ))
for all f ∈ H.
Due to Proposition 7.1 the images of the maps Fl,Fr ,Gl,Gr in H ′ coincide. Letus write H for this space. Moreover, since the maps σ, ν and ρ are isomorphismswe may use any of them to define a unique bornology on H by transferring thebornology from H . We will always view H as a bornological vector space withthis bornology and hence the maps Fl,Fr ,Gl,Gr yield bornological isomorphismsfrom H to H . In particular, the space H satisfies again the approximation property.
We say that a bounded bilinear map b :U ×V → W is nondegenerate if b(u, v)=0 for all u ∈ U implies v = 0 and b(u, v) = 0 for all v ∈ V implies u = 0. SinceH is a regular bornological vector space the canonical pairing between H and H ′
given by 〈 f, ω〉 = ω( f ) is nondegenerate. By construction of the space H there isan obvious injective bounded linear map H → H ′ and we have a nondegeneratepairing between H and H as well. The latter may be extended naturally to apairing between M(H) and H which is again nondegenerate. There are similarconstructions for tensor powers of H and H .
In order to obtain a quantum group structure on H our first aim is to define amultiplication. Consider the transpose map 1∗
: M(H ⊗ H)′ → H ′ of the comul-tiplication given by
1∗(ω)( f )= ω(1( f )).
According to the previous remarks, H ⊗ H can be viewed as a linear subspace ofM(H ⊗ H)′ and 1∗ restricts to a map H ⊗ H → H ′. We shall show that the latteractually yields a bounded linear map H ⊗ H → H . To do this we define a boundedlinear map m : H ⊗ H → H by
m( f ⊗ g)= (id ⊗φ)γ−1l ( f ⊗ g).
Transferring this map according to the isomorphism Fl : H → H we obtain abounded linear map µ : H ⊗ H → H which we call the convolution product. One
116 CHRISTIAN VOIGT
computes
(id ⊗µ)(µ ⊗ id(2))(S ⊗ id(3))(id ⊗1 ⊗ id)(µ(2) ⊗ id)(1 ⊗ id(3))
= (id ⊗µ)(µ ⊗ id(2))(S ⊗ id(3))(id ⊗ ρr ⊗ id)(τ ⊗ id(2)),
and we conclude that
(µ ⊗ id)(S ⊗ id(2))(id ⊗1)µ(2)(1 ⊗ id(2))= (µ ⊗ id)(S ⊗ id(2))(id ⊗ ρr )(τ ⊗ id).
Moreover,
τ(µ ⊗ id)(id ⊗ γ−1l )= (µ ⊗ id)(τ ⊗ id)(id ⊗ S−1
⊗ id)(id ⊗ ρr )(τ ⊗ id).
Using invariance we compute
µ(Fl( f )⊗ Fl(g))(h)= (φ ⊗φ)(µ ⊗ id)(id ⊗ γ−1l )(h ⊗ f ⊗ g)
=1∗(Fl( f )⊗ Fl(g))(h)
and deduce that µ can be identified with 1∗. From this one calculates
µ(µ ⊗ id)(Fl( f )⊗ Fl(g)⊗ Fl(h))(x)= µ(id ⊗ µ)(Fl( f )⊗ Fl(g)⊗ Fl(h))(x),
which means that the convolution product µ is associative. Hence H is a bornolog-ical algebra with convolution as multiplication. According to the above consider-ations we have
(7-1) µ(Fl( f )⊗ Fl(g))= Fl((id ⊗φ)γ−1l ( f ⊗ g))
and an analogous calculation yields the formula
(7-2) µ(Gr ( f )⊗ Gr (g))= Gr ((ψ ⊗ id)ρ−1r ( f ⊗ g))
for the multiplication in H . Actually, this equation can be obtained directly fromthe previous discussion applied to H op cop.
For later use we shall extend the multiplication of H in the following way. From(4-18) and the fact that φ is left invariant we have
(7-3) (id ⊗φ)γr = (id ⊗φ)γlτ(S−1
⊗ id)ρr = (id ⊗φ)(S−1⊗ id)ρr .
Using this observation we define a bounded linear map µl : H ′⊗ H → H ′ by
(7-4) µl(ω⊗ Fl( f ))(x)= (ω ⊗φ)γr (x ⊗ f )= (ω ⊗φ)(S−1⊗ id)ρr (x ⊗ f ).
Inserting ω= ε we see that µl(ω⊗Fl( f ))= 0 for all ω implies f = 0. Conversely,assume µl(ω⊗Fl( f ))= 0 for all f ∈ H . Then we have (ω ⊗φ)γr (h ⊗ f )= 0 forall h, f ∈ H and since γr is an isomorphism this yields ω= 0. Hence µl defines anondegenerate pairing
BORNOLOGICAL QUANTUM GROUPS 117
Similarly, according to (4-17) we have
(7-5) (ψ ⊗ id)ρl = (ψ ⊗ id)(id ⊗ S−1)γ−1r γl = (ψ ⊗ id)(id ⊗ S−1)γl
and we define µr : H ⊗ H ′→ H ′ by
(7-6) µr (Gr ( f )⊗ω)(x)= (ψ ⊗ω)ρl( f ⊗ x)= (ψ ⊗ω)(id ⊗ S−1)γl( f ⊗ x).
As above one sees that the pairing given by µr is nondegenerate. If restricted toH ⊗ H the maps µl and µr are equal to the multiplication map µ. Moreover itis straightforward to check that the maps µl and µr are associative whenever thisassertion makes sense. In the sequel we will simply write µ for the maps µl andµr , respectively.
Using the definition of the modular automorphism σ we obtain
(id ⊗φ)ρr ( f ⊗ x)= (id ⊗φ)γr (x ⊗ σ( f )),
and together with (7-3) this yields the formula
(7-7) µ(ω⊗ Fr ( f ))(x)= (ω ⊗φ)ρr ( f ⊗ x)= (ω ⊗φ)(S ⊗ id)γr ( f ⊗ x)
for the multiplication µ. In a similar way we obtain
(7-8) µ(Gl( f )⊗ω)(x)= (ψ ⊗ω)γl(x ⊗ f )= (ψ ⊗ω)(id ⊗ S)ρl(x ⊗ f )
using (7-5).Our next aim is to show that H is a projective module over itself. In order to do
this we study the regular coaction of H on itself given by γr .
Proposition 7.2. The regular coaction of a bornological quantum group H onitself is a projective H-comodule.
Proof. Choose an element h ∈ H such that φ(h) = 1 and define ν : H → H ⊗ Hby ν( f )= γl(h ⊗ f ). Then we have
(id ⊗φ)ν( f )= (id ⊗φ)γl(h ⊗ f )= φ(h) f = f
for all f ∈ H since φ is left invariant. Hence the map ν satisfies the equation
(7-9) (id ⊗φ)ν = id.
Let us moreover define λ : H → H ⊗ H by λ= γ−1r ν
Now assume that π : V → W is a surjective map of H -comodules with boundedlinear splitting σ and let ξ : H → W be an H -colinear map. We define ζ : H → Vas the composition ζ = (id ⊗φ)ηV (σξ ⊗ id)λ where ηV is the coaction of V .
118 CHRISTIAN VOIGT
We check that ζ is H -colinear. Using Equation (4-17) we obtain
(γ−1r ⊗ id)(γl ⊗ id)(id ⊗ γr )= (id ⊗ S ⊗ id)(ρl ⊗ id)(id ⊗ γr )
= (id ⊗ S ⊗ id)(id ⊗ γr )(id ⊗ S−1⊗ id)(γ−1
r ⊗ id)(γl ⊗ id)
and deduce
(7-10) (λ ⊗ id)γr = (id ⊗ S ⊗ id)(id ⊗ γr )(id ⊗ S−1⊗ id)(λ ⊗ id).
Since S is an algebra and coalgebra antihomomorphism we have τ(S ⊗ S)γr =
ρl(S ⊗ S)τ which yields
(7-11) (S ⊗ id)γr (S−1⊗ id)= τ(S−1
⊗ id)ρl(S ⊗ id)τ.
Using Equations (4-18) and (4-20) we obtain
(7-12) ρr = (S ⊗ id)τγ−1l γr = ρl(S ⊗ id)γr .
Since ηV is right H -linear we calculate
η13V (id ⊗ ρr )(id ⊗µ ⊗ id)= (id ⊗ ρr )η
12V (id ⊗µ ⊗ id)
which yields η13V (id⊗ρr )= (id⊗ρr )η
12V since H is essential. Combining this with
(7-12) implies
(7-13) (id ⊗ ρr )η12V (id ⊗ γ−1
r )= η13V (id ⊗ ρl)(id ⊗ S ⊗ id).
Using Equations (7-10), (7-11), (7-13) and (7-3) we calculate (ζ ⊗ id)γr = ηV (ζ ⊗
id) which shows that ζ is H -colinearWe now prove that ζ is a lifting for ξ . Since π is colinear, the diagram
H ⊗ Hσξ⊗id //
id��
V ⊗ HηV //
π⊗id��
V ⊗ Hid⊗φ //
π⊗id��
V
π
��H ⊗ H
ξ⊗id // W ⊗ HηW // W ⊗ H
id⊗φ // W
is commutative, and moreover ηW (ξ ⊗ id) = (ξ ⊗ id)γr because ξ is colinear. Asa consequence we get πζ = ξ since
(id ⊗φ)γrλ= (id ⊗φ)ν = id
by the definition of λ. �
Consider the transposed right regular coaction ρ of H on itself given by ρ =
γ−1l τ . Since Equation (4-19) for H op yields
(7-14) γr (S−1⊗ id)= (S−1
⊗ id)γ−1l τ
BORNOLOGICAL QUANTUM GROUPS 119
we see that the transposed right regular coaction ρ corresponds to the right regularcoaction γr under the linear automorphism of H given by S−1. It follows thatthe map ρ is indeed a coaction and that the coactions ρ and γr yield isomorphiccomodules. Hence the comodule defined by ρ is projective due to Proposition 7.2
As above let m : H ⊗ H → H be the map corresponding to the multiplication ofH under the isomorphism Fl . By the definition of the right regular coaction we get
(7-15) m = (id ⊗φ)ρτ.
The pentagon relation for the operator γ copr = τγl = ρ−1 can be written as
(7-16) ρ23ρ13ρ12= ρ12ρ23
;
together with the formula (id ⊗φ)τρ−1= (id ⊗φ)γl = (id ⊗φ)τ this implies
(m ⊗ id)ρ13= (id ⊗ id ⊗φ)(id ⊗ τ)(ρτ ⊗ id)ρ13,= ρ(m ⊗ id)
which means that the map m is right H -colinear if we view H ⊗ H as a right H -comodule using the coaction ρ13. Since m has a bounded linear splitting we obtaina colinear splitting σ : H → H ⊗ H due to Proposition 7.2. In other words, wehave (σ ⊗ id)ρ = ρ13(σ ⊗ id), which yields
σm = (id(2) ⊗φ)(σ ⊗ id)ρτ = (id ⊗φ ⊗ id)(ρτ ⊗ id)(id ⊗σ)= (m ⊗ id)(id ⊗σ).
Translating this to H using the isomorphism Fl we see that there is a H -linearsplitting for the multiplication map µ if H acts by multiplication on the left tensorfactor of H ⊗ H . Using such a splitting it is straightforward to check that H is anessential bornological algebra.
We define a linear form ψ on H by ψ(Fl( f ))= ε( f ) and compute
ψ(Fl( f )Fl(g))= (ε ⊗φ)γ−1l ( f ⊗ g)= φ(S−1(g) f )
for all f, g ∈ H which implies that ψ is faithful since φ is faithful. Hence thealgebra H is equipped with a faithful bounded linear functional. We will see belowthat ψ is right invariant for the comultiplication of H , but first, of course, we haveto construct this comultiplication.
To do this we define a bounded linear map γr : H ⊗ H → H ⊗ H by
γr (Fl( f )⊗ Fl(g))= (Fl ⊗ Fl)τγ−1l ( f ⊗ g).
It is evident that γr is a bornological isomorphism. Let us show that γr commuteswith right multiplication on the second tensor factor. Using the pentagon relation(7-16) for ρ and (φ ⊗ id)ρ = φ ⊗ id we have
(id ⊗ id ⊗φ)(τ ⊗ id)(γ−1l ⊗ id)(id ⊗ γ−1
l )( f ⊗ g ⊗ h)
= (id ⊗ id ⊗φ)(id ⊗ γ−1l )(τ ⊗ id)(γ−1
l ⊗ id)( f ⊗ g ⊗ h),
120 CHRISTIAN VOIGT
and translating this using the map Fl we obtain
(7-17) γr (id ⊗ µ)= (id ⊗ µ)(γr ⊗ id),
as desired. Similarly, we define a bornological automorphism ρl of H ⊗ H by
ρl(Gr (g)⊗ Gr ( f ))= (Gr ⊗ Gr )τρ−1r (g ⊗ f ).
and using formula (7-2) we obtain
(7-18) ρl(µ ⊗ id)= (µ ⊗ id)(id ⊗ ρl).
From (7-3) and (7-5) we have
(7-19) γr (ω⊗Fl(g))(x ⊗ y)= (ω⊗φ)(µ⊗ id)(id⊗S−1⊗ id)(id⊗ρr )(x ⊗ y⊗g)
and
(7-20) ρl(Gr ( f )⊗ω)(x ⊗ y)= (ψ ⊗ω)(id⊗µ)(id⊗S−1⊗id)(γl ⊗id)( f ⊗x ⊗ y)
for all ω ∈ H . Using (7-4) and (7-6) we calculate
(id ⊗ µ)(ρl ⊗ id)(Gr ( f )⊗ω⊗ Fl(g))(x ⊗ y)
= (µ ⊗ id)(id ⊗ γr )(Gr ( f )⊗ω⊗ Fl(g))(x ⊗ y),
which yields
(7-21) (id ⊗ µ)(ρl ⊗ id)= (µ ⊗ id)(id ⊗ γr ).
Using the pentagon relation (7-16) for ρ we compute
γr (µ ⊗ id)(Fl( f )⊗ Fl(g)⊗ Fl(h))
= (µ ⊗ id)(τ ⊗ id)(id ⊗ γr )(τ ⊗ id)(id ⊗ γr )(Fl( f )⊗ Fl(g)⊗ Fl(h)),
which gives
(7-22) γr (µ ⊗ id)= (µ ⊗ id)(τ ⊗ id)(id ⊗ γr )(τ ⊗ id)(id ⊗ γr ).
A similar computation shows that
(7-23) ρl(id ⊗ µ)= (id ⊗ µ)(id ⊗ τ)(ρl ⊗ id)(id ⊗ τ)(ρl ⊗ id).
Finally, we have
(ρl ⊗ id)(id ⊗ γr )(Gr ( f )⊗ω⊗ Fl(g))(x ⊗ y ⊗ z)
= (id ⊗ γr )(ρl ⊗ id)(Gr ( f )⊗ω⊗ Fl(g))(x ⊗ y ⊗ z),
according to (7-19) and (7-20), and hence
(7-24) (ρl ⊗ id)(id ⊗ γr )= (id ⊗ γr )(ρl ⊗ id).
BORNOLOGICAL QUANTUM GROUPS 121
Using the properties of the maps γr and ρl obtained so far we shall construct thecomultiplication for H according to the following general result.
Proposition 7.3. Let K be an essential bornological algebra satisfying the ap-proximation property equipped with a faithful bounded linear functional. If γr andρl are bornological automorphisms of K ⊗ K such that
(a) γr (id ⊗µ)= (id ⊗µ)(γr ⊗ id),
(b) γr (µ ⊗ id)= (µ ⊗ id)(τ ⊗ id)(id ⊗ γr )(τ ⊗ id)(id ⊗ γr ),
(c) ρl(µ ⊗ id)= (µ ⊗ id)(id ⊗ ρl),
(d) ρl(id ⊗µ)= (id ⊗µ)(id ⊗ τ)(ρl ⊗ id)(id ⊗ τ)(ρl ⊗ id),
(e) (id ⊗µ)(ρl ⊗ id)= (µ ⊗ id)(id ⊗ γr ),
(f) (ρl ⊗ id)(id ⊗ γr )= (id ⊗ γr )(ρl ⊗ id),
then there exists a unique comultiplication 1 : K → M(K ⊗ K ) such that γr andρl are the associated Galois maps.
If there exist also bornological automorphisms γl and ρr of K ⊗ K such that
(g) (µ ⊗ id)(id ⊗ τ)(γl ⊗ id)= γl(id ⊗µ),
(h) (id ⊗µ)(γl ⊗ id)= (µ ⊗ id)(id ⊗ τ)(γr ⊗ id)(id ⊗ τ),
(i) (id ⊗µ)(τ ⊗ id)(id ⊗ ρr )= ρr (µ ⊗ id),
(j) (µ ⊗ id)(id ⊗ ρr )= (id ⊗µ)(τ ⊗ id)(id ⊗ ρl)(τ ⊗ id),
then these maps are the remaining Galois maps. Thus all Galois maps are isomor-phisms in this case.
Proof. Using condition a) it is straightforward to check that
µ(2)(1l ⊗ id(2))= (µ ⊗ id)(id ⊗ τ)(γr ⊗ id)(id ⊗ τ)
defines a bounded linear map 1l : K → Ml(K ⊗ K ). According to condition b)the map 1l is actually a homomorphism. Similarly,
µ(2)(id(2) ⊗1r )= (id ⊗µ)(τ ⊗ id)(id ⊗ ρl)(τ ⊗ id)
defines a homomorphism 1r : K → Mr (K ⊗ K ) due to conditions c) and d).Condition e) ensures that these maps combine to an algebra homomorphism 1 :
K → M(K ⊗K ). It is straightforward to show that γr and ρl are the correspondingGalois maps. Moreover 1 is uniquely determined by these maps
We have to prove that the homomorphism 1 is essential. Let us show that thenatural map K ⊗K (K ⊗K )→ K ⊗K is an isomorphism where the module structureon K ⊗ K is given by 1. Since K is essential we have K ⊗K K ∼= K and we can
122 CHRISTIAN VOIGT
identify the source of the previous map with K ⊗K (K ⊗ (K ⊗K K )) in a naturalway. It is easy to check that γ 13
r descends to a bounded linear map
ξ : K ⊗K (K ⊗ (K ⊗K K ))→ (K ⊗K K ) ⊗ (K ⊗K K )
and the composition of ξ withµ⊗µ can be identified with the map we are interestedin. Hence it suffices to show that ξ is an isomorphism. Consider the maps
p = µ ⊗ id ⊗µ ⊗ id − (id ⊗µ ⊗ id(2))γ 24r (id(4) ⊗µ),
q = µ ⊗ id ⊗µ ⊗ id − id ⊗µ ⊗ id ⊗µ,
defined on the six-fold tensor product of K with itself. The source and target ofξ are the quotients of K ⊗4 by the closure of the image of p and q , respectively.Using conditions a) and b) it is straightforward to verify the relation qκ = γ 13
r pwhere κ is the bornological automorphism of K ⊗6 defined by
κ = (id(2) ⊗ τ ⊗ id(2))γ 13r γ 23
r (id(2) ⊗ τ ⊗ id(2)).
This relation shows that ξ is actually a bornological isomorphism. Using the mapρl one proves in a similar way that (K ⊗ K ) ⊗K K → K ⊗ K is an isomorphism.We conclude that 1 is essential. Having established this, condition f) immediatelyyields that 1 is coassociative. Hence 1 is a comultiplication.
If there exists maps γl and ρr with the properties stated in conditions g) and h)then these maps describe the remaining Galois maps associated to 1. It follows inparticular that all Galois maps yield isomorphisms from K ⊗ K into itself in thiscase. �
We have already shown above that the maps γr and ρl satisfy the assumptions ofProposition 7.3. Let us write 1 for the comultiplication on H defined in this way.By construction, the Galois maps γr and ρl associated to 1 are isomorphisms.
To treat the remaining Galois maps for 1, we define abstractly
γl(Fr ( f )⊗ Fr (g))= (Fr ⊗ Fr )τρ−1l ( f ⊗ g),
ρr (Gl(g)⊗ Gl( f ))= (Gl ⊗ Gl)τγ−1r (g ⊗ f ).
Applying the preceding discussion to H op we see that γl and ρr satisfy conditions(g) and (i) in Proposition 7.3. Using (7-3) and (7-5) it is straightforward to obtainthe formulas
(7-25) γl(ω⊗ Fr (g))(x ⊗ y)= (ω ⊗φ)(µ ⊗ id)(id ⊗ τ)(ρr ⊗ id)(g ⊗ x ⊗ y)
and
(7-26) ρr (Gl( f )⊗ω)(x ⊗ y)= (ψ ⊗ω)(id ⊗µ)(τ ⊗ id)(id ⊗ γl)(x ⊗ y ⊗ f )
BORNOLOGICAL QUANTUM GROUPS 123
for the maps γl and ρr . Using the definition of µ and Equations (7-25), (7-3) and(7-19), we compute
(id ⊗ µ)(γl ⊗ id)(ω⊗ Fr ( f )⊗ Fl(g))(x ⊗ y)
= (γl(ω⊗ Fr ( f )) ⊗φ)(x ⊗ γr (y ⊗ g))
= (µ ⊗ id)(id ⊗ τ)(γr ⊗ id)(id ⊗ τ)(ω⊗ Fr ( f )⊗ Fl(g))(x ⊗ y),
which yields condition (h). Using Equations (7-26), (7-5) and (7-20) one obtainscondition (j) in a similar way. Hence it follows from Proposition 7.3 that all Galoismaps associated to 1 are isomorphisms.
It remains to exhibit the Haar functionals for the comultiplication 1. The proofof the following proposition is straightforward.
Proposition 7.4. Let H be a bornological quantum group. Then the linear form ψ
on H defined byψ(Fl( f ))= ε( f )
is a faithful right invariant functional on H . Similarly, the linear form φ given by
φ(Gr ( f ))= ε( f )
is a faithful left invariant functional on H .
We have now completed to proof of the following theorem.
Theorem 7.5. Let H be a bornological quantum group. Then H with the structuremaps described above is again a bornological quantum group.
The bornological quantum group H will be called the dual quantum group of H .It is instructive to describe explicitly the counit and the antipode of H . Considerthe map ε : H → C given by
ε(ω)= ω(1),
where H is viewed as a subspace of M(H)′ according to the nondegenerate pairingH × M(H)→ C. The explicit formulas
ε(Fl( f ))= φ( f ), ε(Fr ( f ))= φ( f ), ε(Gl( f ))=ψ( f ), ε(Gr ( f ))=ψ( f )
show that the map ε is bounded and nonzero. It is straightforward to check that εis an algebra homomorphism and we calculate
(ε ⊗ id)γr (Fl( f )⊗ Fl(h))= (Fl ⊗φ)γ−1l ( f ⊗ h)= µ(Fl( f )⊗ Fl(h)),
(id ⊗ ε)ρl(Gr (h)⊗ Gr ( f ))= (ψ ⊗ Gr )ρ−1r (h ⊗ f )= µ(Gr (h)⊗ Gr ( f )),
which shows (ε ⊗ id)γr = µ and (id ⊗ ε)ρl = µ. One can then proceed as in theproof of Theorem 4.6 to show that ε is nondegenerate. By the uniqueness assertionof Theorem 4.6 we see that the map ε is indeed the counit for H .
124 CHRISTIAN VOIGT
Similarly, we define S : H → H by S(ω)( f ) = ω(S( f )), and using ψ = S(φ)we obtain the formulas
S(Fl( f ))= Gr (S−1( f )), S(Fr ( f ))= Gl(S−1( f )).
It follows that S is a bounded linear automorphism of H . Using (7-3) we compute
µ(S ⊗ id)(Fl( f )⊗ Fl(g))(x)= (ε ⊗ id)γ−1r (Fl( f )⊗ Fl(g))(x),
which shows µ(S ⊗ id) = (ε ⊗ id)γ−1r . In a similar way one obtains the relation
µ(id ⊗ S)= (id ⊗ ε)ρ−1l . Inspecting the constructions in the proof of Theorem 4.6
we see that S is the antipode of H .
Theorem 7.6 (Pontrjagin duality theorem). Let H be a bornological quantumgroup. Then the double dual quantum group of H is canonically isomorphic toH.
Proof. We define a linear map P : H → (H)′ by P( f )(ω) = ω( f ) for all f ∈ Hand ω ∈ H . Using Proposition 7.4 we compute
(GlFl( f ))(Fl(h))= ψ(Fl(h)Fl( f ))= ψ(Fl ⊗φ)γ−1l ( f ⊗ h)
= (ε ⊗φ)γ−1l ( f ⊗ h)= φ(S−1( f )h)= Fl(h)(S−1( f ))
for all f, g ∈ H , where Gl is the map Gl for H . This implies
P( f )= GlFl(S( f ))
and shows that P defines a bornological isomorphism from H to ˆH . In a similarway one has P( f )= Fr Gr (S( f )). Further, using ψ = S−1(φ) one calculates
(Fl SGl( f ))(Gr (h))= φ(Gr (h)Fr (S−1( f )))
= (φ ⊗ ε)ρ−1r (h ⊗ S−1( f ))= Gr (h)( f ),
which shows P = Fl SGl .Next consider the transpose µ∗
: H ′→ (H ⊗ H)′ of the multiplication map,
which is given by
µ∗(ω)( f ⊗ g)= ωµ( f ⊗ g)= ω( f g)
for all f, g ∈ H . In particular, we obtain a bounded linear map µ∗: H → (H ⊗ H)′
by restriction. Using the isomorphism P we can view 1 as a map from H → (H ⊗
H)′ as well. Equivalently, we have bounded linear maps from H into Hom(H, H ′)
given byµ∗(ω)(g)( f )= ωµ( f ⊗ g)
BORNOLOGICAL QUANTUM GROUPS 125
and likewise for 1. Using (7-14) we calculate
1(Fl(h))(g)= (id ⊗ ψ)γr (Fl(h)⊗ Fl S(g))= Fl(gh)
and thus obtain, using the definition of S−1,
1(Fl(h))(g)( f )= ψ(Fl(gh)Fl S( f ))= ψ(Fl ⊗φ)γ−1l (gh ⊗ S( f ))
= (ε ⊗φ)γ−1l (gh ⊗ S( f ))= φ( f gh)= µ∗(Fl(h))(g)( f ),
which shows that 1 can be identified with the transpose µ∗ of the multiplication.Similarly, we have seen in the constructions above that µ can be identified withthe transpose 1∗ of the comultiplication.
With this in mind it is easy to check that P is an algebra homomorphism and acoalgebra homomorphism. Hence P is an isomorphism of bornological quantumgroups. �
8. Duality for modules and comodules
In this section we study the duality between essential modules and comodules overa bornological quantum group and its dual
Let H be a bornological quantum group and let η : V → HomH (H, V ⊗ H) bean essential H -comodule. We define a bounded linear map D(η) : H ⊗ V → V by
D(η)(Fl( f )⊗ v)= (id ⊗φ)η(v⊗ f ).
For later use we need another description of this map. Since H is an essentialalgebra we may view η as a bounded linear map from V into HomH (H, V ⊗ H)∼=HomH (H ⊗H H, V ⊗ H). Under the latter isomorphism η(v) corresponds to themap (id ⊗µ)(η(v) ⊗ id). Moreover, using notation and results from Section 5 wehave
φ(hgν( f ))= ψ(hg f )= φ(hν(g f ))
for all f, g, h ∈ H which implies that ν is left H -linear. Together with the relation
(id ⊗φ)(id ⊗µ)(η(v)(g)⊗ ν( f ))= (id ⊗ψ)(id ⊗µ)(η(v)(g)⊗ f )
we thus obtain D(η)(Gl( f )⊗ v)= (id ⊗ψ)η(v⊗ f ). Next we compute
D(η)(id ⊗ D(η))(Fl( f )⊗ Fl(g)⊗ v)
= (id ⊗φ)η(D(η)(Fl(g)⊗ v)⊗ f )
= (id ⊗φ ⊗φ)(id ⊗ γr )η12(id ⊗ γ−1r )(v⊗ f ⊗ g).
From (7-3) we have (φ ⊗φ)γr = (φ ⊗φ)(S−1⊗ id)ρr , and using (4-18) and (7-13)
we obtain
D(η)(id ⊗ D(η))(Fl( f )⊗ Fl(g)⊗ v)= D(η)(µ ⊗ id)(Fl( f )⊗ Fl(g)⊗ v),
126 CHRISTIAN VOIGT
which shows that V becomes a left H -module in this way.We want to show that V is actually an essential H -module. In order to do this it
is convenient to work with the map (id ⊗φ)η instead of D(η). There is an evidentbounded linear splitting σ : V → V ⊗ H of this map given by σ(v)= η−1(v⊗ h)where h is chosen such that φ(h)= 1. If we identify H ⊗H V accordingly with aquotient Q of V ⊗ H we have the relation
(id ⊗ id ⊗φ)η13 = (id ⊗ id ⊗φ)(id ⊗ γ−1l )
in this quotient. Now we see, as in the proof of Proposition 7.2 and using formula(7-14), that
(id ⊗ id ⊗φ)η12η13 = (id ⊗φ ⊗ id)η12(id ⊗ γ−1l ),
which implies that
η−1(id ⊗φ ⊗ id)η12 = (id ⊗ id ⊗φ)η13(id ⊗ γl)= id ⊗ id ⊗φ
in Q. It follows that σ(id⊗φ)η is the identity map on Q. Translating this back toH ⊗H V we deduce that V is an essential module.
An H -colinear map f : V → W is easily seen to be H -linear for the modulestructures defined in this way. Hence we have proved the following statement.
Proposition 8.1. Let H be a bornological quantum group and let H be the dualquantum group. The previous construction defines a functor D from Comod-H toH-Mod.
Conversely, let λ : H ⊗ V → V be an essential left H -module. By slight abuseof notation we write λ−1 for the inverse of the isomorphism H ⊗H V ∼= V inducedby λ. We define a bounded linear map D(λ) : V ⊗ H → V ⊗ H by
D(λ)(v⊗ Fl( f ))= (id ⊗ Fl)τ (id ⊗ λ)(γ−1l τ ⊗ id)(id ⊗ λ−1)( f ⊗ v),
which is seen to be well-defined since γ−1l τ is right H -linear for the action by mul-
tiplication on the second tensor factor. It is evident that D(λ) is an isomorphism.Since λ is left H -linear we calculate, with ρ = γ−1
l τ ,
(id ⊗ λ)(γ−1l τ ⊗ id)(id ⊗ λ−1)(id ⊗φ ⊗ id)(γ−1
l ⊗ id)( f ⊗ g ⊗ v)
= (id ⊗φ ⊗ id)(γ−1l ⊗ id)(τ ⊗ λ)(id ⊗ γ−1
l τ ⊗ id)(τ ⊗ λ−1)( f ⊗ g ⊗ v),
using (φ ⊗ id) = (φ ⊗ id)ρ as well as the pentagon relation (7-16) for the map ρ.This shows that
D(λ)(id ⊗ µ)(v⊗ Fl( f )⊗ Fl(g))= (id ⊗ µ)(D(λ) ⊗ id)(v⊗ Fl( f )⊗ Fl(g)),
BORNOLOGICAL QUANTUM GROUPS 127
which means that D(λ) is right H -linear. Again by the pentagon relation for ρ andthe H -linearity of λ is we have
(τ ⊗λ)(id⊗γ−1l τ ⊗id)(τ ⊗id(2))(id⊗γ−1
l τ ⊗id)(id(2)⊗λ−1)(τγ−1l ⊗id)( f ⊗g⊗v)
= (τγ−1l ⊗id)(τ ⊗id)(id(2)⊗λ)(id⊗γ−1
l τ ⊗id)(τ ⊗id(2))(id(2)⊗λ−1)( f ⊗g⊗v),
which shows that
D(λ)12 D(λ)13(id ⊗ γr )(v⊗ F( f )⊗ F(g))= (id ⊗ γr )D(λ)12(v⊗ F( f )⊗ F(g)).
Hence D(λ) is a right coaction of H on V . It is easy to check that an H -equivariantmap f : V → W between H -modules defines an H -colinear map between theassociated comodules.
Proposition 8.2. Let H be a bornological quantum group and let H be the dualquantum group. There is a natural functor from H-Mod to Comod-H which willagain be denoted by D.
Theorem 8.3 (Duality theorem for modules and comodules). Let H be a borno-logical quantum group. Every essential left H-module is an essential right H -comodule in a natural way and vice versa. This yields inverse isomorphismsbetween the category of essential H-modules and the category of essential H -comodules. These isomorphisms are compatible with tensor products.
Of course an analogous statement holds for right modules and left comodules.
Proof. We check that the functors defined above are inverse to each other if wetake into account the Pontrjagin duality Theorem 7.6. By Equation (4-20) we haveγ−1
l = τ(S−1⊗ id)ρl(S ⊗ id) and hence
(ε ⊗ id)γ−1l τ(S ⊗ id)= (id ⊗ ε)(S−1
⊗ id)ρl(S ⊗ S)τ = S−1µ(S ⊗ S)τ = µ.
Using the definition of the right Haar functional ψ on H we thus compute for anessential H -module λ : H ⊗ V → V
(id ⊗ ψ)D(λ)(v⊗ Fl(S( f )))= λ(id ⊗ λ)(id ⊗ λ−1)( f ⊗ v)= λ( f ⊗ v).
Consequently we have
DD(λ)(GlFl(S( f ))⊗ v)= λ( f ⊗ v)
and according to Pontrjagin duality this shows that the module structure DD(λ)can be identified with λ
Conversely, let η : V → HomH (V, V ⊗ H) be an essential H -comodule. Using(7-14) for H we compute
DD(η)(v⊗Fl S(ω))= (id⊗Fl)τ (id⊗ D(η))(γ−1l τ ⊗id)(id⊗ D(η)−1)(S(ω)⊗v)
= τ(Fl S ⊗id)(id⊗ D(η))(γr ⊗id)(id⊗ D(η)−1)(ω⊗v),
128 CHRISTIAN VOIGT
and thus obtain
DD(η)(D(η)(Fl(g)⊗ v)⊗ Fl SGl( f ))= (id ⊗ Fl SGl)η(D(η)(Fl(g)⊗ v)⊗ f ),
which implies
DD(η)(v⊗ Fl SGl( f ))= (id ⊗ Fl SGl)η(v⊗ f )
since D(η) is an essential H -module. Again by Pontrjagin duality this shows thatDD(η) is isomorphic to η
Consider H ⊗H (H ⊗ (H ⊗H H)) and (H ⊗H H) ⊗ (H ⊗H H) as H -modulesby multiplication on the first tensor factor and by the diagonal action on the firstand third tensor factors, respectively. Then the isomorphism ξ used in the proofof Proposition 7.3 is H -linear. Using this it is straightforward to check that thefunctor D from H-Mod to Comod-H is compatible with tensor products. �
We now use this duality result to construct the dual of a morphism betweenbornological quantum groups.
Proposition 8.4. Let α : H → M(K ) be a morphism of bornological quantumgroups. Then there exists a unique morphism α : K → M(H) such that
〈α( f ), ω〉 = 〈 f, α(ω)〉
for all f ∈ H and ω ∈ K .
Proof. Uniqueness of α follows immediately from the nondegeneracy of the pairingbetween H and M(H). Consider the transposed right regular coaction ρ = γ−1
l τ
on H . The dual action of the pushforward coaction α∗(ρ) yields a left K -modulestructure on H . Using the linear isomorphism Fl we may view this as a K -modulestructure on H . Associativity of the multiplication in H and (7-15) shows that weobtain in fact a bounded linear map αl : K → Ml(H). Similarly, the map γ = ρ−1
r τ
defines a left coaction of H on itself, and the dual action of the correspondingpushforward coaction determines a right K -module structure on H . This actionyields a homomorphism αr : K → M(H). Using Lemma 4.5 for H cop we obtain
(id ⊗ ρ)(γ ⊗ id)= (γ ⊗ id)(id ⊗ ρ),
so the resulting left and right K -module structures on H commute. Hence the mapsαl and αr yield a nondegenerate homomorphism α : K → M(H).
Consider also the transpose α∗: K → H ′ of α given by α∗(ω)( f ) = ω(α( f )).
Then we have 〈 f, α∗(ω)〉 = 〈α( f ), ω〉. Moreover the calculation after (4-16) forH cop gives
(µ ⊗ id)(id ⊗ ρ)= (id ⊗µ)(id ⊗ S−1⊗ id)(τρr ⊗ id).
BORNOLOGICAL QUANTUM GROUPS 129
Using (7-3), the definition of αl an that of µ : H ′⊗ H → H ′ we calculate
µ(αl ⊗ id)(Fl(k)⊗ Fl( f ))(h)= µ(α∗⊗ id)(Fl(k) ⊗ Fl( f ))(h),
where λl denotes the isomorphism H ⊗H K ∼= K induced by α. This shows
µ(αl(Fl(k))⊗ Fl( f ))= µ(α∗(Fl(k))⊗ Fl( f ))
for all k ∈ K and f ∈ H . Similarly we have
µ(Gr ( f ))⊗ αr (Gr (k)))= µ(Gr ( f ))⊗α∗(Gr (k)))
and we obtain α(ω)= α∗(ω) for all ω ∈ KWe shall only sketch how to show that α is a coalgebra homomorphism. Using
(7-3) one obtains
(φ ⊗φ)µ(2)(ρl ⊗ id(2))= (φ ⊗φ)(µ ⊗ id)(id ⊗µ ⊗ id)γ 24r
= (φ ⊗φ)µ(2)(id(2) ⊗ τγ−1l ),
which shows that
〈ρl( f ⊗ g),Fl(h)⊗ Fl(k)〉 = 〈 f ⊗ g, γr (Fl(h)⊗ Fl(k))〉
for all f, g, h, k ∈ H . This relation extends to the case where f and g are multipliersof H and we have similar statements involving other Galois maps. Based on thiswe calculate
〈 f ⊗ g, (α ⊗ α)γr (Fl(k)⊗ Fl(l))〉 = 〈 f ⊗ g, γr (α ⊗ α)(Fl(k)⊗ Fl(l))〉
and deduce (α ⊗ α)γr = γr (α ⊗ α), which easily implies that α is compatible withthe comultiplication. �
9. Bornological quantum groups associated to Lie groups
In this section we describe a dual pair of bornological quantum groups associatednaturally to every Lie group. These bornological quantum groups are generaliza-tions of the Hopf algebra of functions C(G) and the group algebra CG of a finitegroup G. We will indicate at the end of this section how the constructions describedbelow can be extended to arbitrary locally compact groups
If M is a smooth manifold we let D(M) be the space of smooth functions on Mwith compact support. The space D(M) is equipped with the bornology associatedto its natural LF-topology. The following assertion is immediate.
Lemma 9.1. Let M be a smooth manifold. The multiplier algebra of the algebraD(M) of smooth functions with compact support with pointwise multiplication isthe algebra E(M) of all smooth functions.
130 CHRISTIAN VOIGT
Now let G be a Lie group. We choose a left Haar measure dt and denote themodular function of G by δ. Let us write C∞
c (G) for the bornological algebraof smooth functions on G with pointwise multiplication. Using Lemma 9.1 onedefines the comultiplication 1 : C∞
c (G)→ M(C∞c (G × G)) by
1( f )(r, s)= f (rs).
Proposition 9.2. Let G be a Lie group. Then the algebra C∞c (G) of smooth func-
tions with compact support on G is a bornological quantum group.
Proof. It is straightforward to check that all Galois maps associated to 1 are iso-morphisms. A left invariant integral φ for C∞
c (G) is given by integration withrespect to the Haar measure. �
The counit ε : C∞c (G)→ C is given by ε( f )= f (e) where e is the unit element
of G. The antipode S : C∞c (G) → C∞
c (G) is defined by S( f )(t) = f (t−1). Themodular element in M(C∞
c (G)) is given by the modular function δ.We describe the dual of C∞
c (G). We write D(G) for this bornological quantumgroup and refer to it as the smooth group algebra of G. The underlying bornologicalvector space is of course again the space of smooth functions with compact supporton G. Multiplication is given by the convolution product
( f ∗ g)(t)=
∫G
f (s)g(s−1t) ds
which turns D(G) into a bornological algebra. Note that D(G) does not have aunit unless G is discrete. The corresponding multiplier algebra is determined in[Meyer 2004b].
Proposition 9.3. Let G be a Lie group. The multiplier algebra of the smooth groupalgebra D(G) is the algebra E′(G) of distributions on G with compact support.
We remark that the complex group ring CG is contained in M(D(G))= E′(G)as the subalgebra spanned by the Dirac distributions δs for s ∈ G.
Using Proposition 9.3 one describes the comultiplication1 :D(G)→E′(G×G)by
1( f )(h)=
∫G
f (s)h(s, s) ds.
The counit ε : D(G)→ C is defined by
ε( f )=
∫G
f (s) ds,
and the antipode S : D(G)→ D(G) is given by S( f )(t)= δ(t) f (t−1). The generaltheory developed in the previous sections yields immediately the following result.
Proposition 9.4. Let G be a Lie group. Then the smooth group algebra D(G) ofG is a bornological quantum group.
BORNOLOGICAL QUANTUM GROUPS 131
A left and right invariant integral φ for D(G) is given by φ( f )= f (e) where eis the identity element
As mentioned above, one may as well consider smooth functions on arbitrarylocally compact groups G and obtain corresponding bornological quantum groupsC∞
c (G) and D(G). The definition of the space of smooth functions in this settinginvolves the structure theory of locally compact groups. More information can befound in [Meyer 2004b] where smooth representations of locally compact groupson bornological vector spaces are studied
Actually, it is immediate from the definitions that a smooth representation ofthe group G is the same thing as an essential comodule over C∞
c (G). In [Meyer2004b] it is shown that the category of smooth representations of G is naturallyisomorphic to the category of essential modules over D(G). This result is a specialcase of Theorem 8.3 and explains the motivation for the general definitions ofessential modules and comodules in Section 6.
10. Schwartz algebras and discrete groups
In this section we describe bornological quantum groups arising from Schwartzalgebras of certain Lie groups as well as from algebras of functions satisfyingvarious decay conditions on finitely generated discrete groups.
We begin with the abelian Lie group G = Rn . Let S(Rn) be the Schwartz spaceof rapidly decreasing smooth functions on Rn . The topology of this nuclear Frechetspace is defined by the seminorms
pkα( f )= sup
x∈Rn
∣∣∣∣∂α f (x)∂xα
(1 + |x |)k∣∣∣∣
for any multiindex α and any nonnegative integer k. Here |x | denotes the euclideannorm of x . The space S(Rn) becomes an essential bornological algebra with thepointwise multiplication of functions. In order to identify the corresponding mul-tiplier algebra recall that a function f ∈ C∞(Rn) is called slowly increasing if forevery multiindex α there exists an integer k such that
supx∈Rn
∣∣∣∣ 1(1 + |x |)k
∂α f (x)∂xα
∣∣∣∣<∞.
Slowly increasing functions on Rn form an algebra under pointwise multiplication.In fact:
Lemma 10.1. The multiplier algebra M(S(Rn)) is the algebra of slowly increas-ing functions on Rn .
To define the quantum group structure of S(Rn) the formulas for C∞c (R
n) carryover.
132 CHRISTIAN VOIGT
Proposition 10.2. The algebra S(Rn) of rapidly decreasing functions on Rn is abornological quantum group.
We describe also the dual of S(Rn). We will denote this quantum group byS∗(Rn) and call it the tempered group algebra of Rn . The underlying algebrastructure is given by S(Rn) with convolution multiplication. In order to determinethe multiplier algebra of S∗(Rn) let us denote by B(Rn) the space of all smoothfunctions f on Rn such that all derivatives of f are bounded. The topology onB(Rn) is given by uniform convergence of all derivatives. By definition, a boundeddistribution is a continuous linear form on the space B(Rn). A distribution T ∈
D′(Rn) has rapid decay if the distribution T (1 + |x |)k is bounded for all k ≥ 0.Using Fourier transform on obtains the following statement.
Lemma 10.3. The multiplier algebra M(S(Rn)) of S∗(Rn) is the algebra of dis-tributions with rapid decay.
The comultiplication, counit, antipode and the Haar integral for S∗(Rn) can bedetermined in the same way as for the smooth group algebra D(Rn). Remark thatthe classical Fourier transform can be viewed as an isomorphism of bornologicalquantum groups S∗(Rn)∼= S(Rn) where Rn is the dual group of Rn .
The tempered group algebra S∗(R) and its dual as well as corresponding crossedproducts have been considered by Elliot, Natsume and Nest in [Elliott et al. 1988].
We remark that the abelian case treated above can be extended easily to simplyconnected nilpotent Lie groups. The algebra S(G) of Schwartz functions on anilpotent Lie group G has been considered by Natsume and Nest [1994] in con-nection with their study of the cyclic cohomology of the Heisenberg group.
Now let 0 be a finitely generated discrete group equipped with a word metric.We denote by L the associated length function on 0. The function L satisfies
L(e)= 0, L(t)= L(t−1) and L(st)≤ L(s)+ L(t)
for all s, t ∈ 0. Following [Meyer 2006] we define several function spaces associ-ated to 0. For every k ∈ R consider the norm
pk( f )=
∑t∈0
| f (t)|(1 + L(t))k
on the complex group ring C0 and denote by Sk(0) the corresponding Banachspace completion. We write also l1(0) instead of S0(0). Moreover let S(0) bethe completion of C0 with respect to the family of norms pk for all k ∈ N. Thenatural map Sk+1(0)→ Sk(0) is compact for all k ∈ N and hence S(0) is a FrechetSchwartz space. We call S(0) the space of Schwartz functions on 0.
BORNOLOGICAL QUANTUM GROUPS 133
Consider moreover the norm
pα( f )=
∑t∈0
| f (t)|αL(t)
for α > 1. We write l1(0, α) for the completion of C0 with respect to this normand O(0) for the completion with respect to the family pn for n ∈ N. Moreoverlet Sω(0) be the direct limit of the Banach spaces l1(0, α) for α > 1. All thesefunction spaces do not depend on the choice of the word metric.
All function spaces considered above become bornological algebras with theconvolution product. The comultiplication, counit, antipode and the Haar func-tional of C0 extend continuously to the completions.
Proposition 10.4. Let 0 be a finitely generated discrete group. Then the algebrasl1(0),S(0),O(0) and Sω(0) are bornological quantum groups in a natural way.
The algebra structure of the corresponding dual quantum groups is obtained byequipping the above spaces of functions with pointwise multiplication.
11. Rieffel deformation
Rieffel [1993a] studied deformation quantization for Poisson brackets arising fromactions of Rd . Although the main focus in that monograph is on the study of theC∗-algebras arising in this way, a large part of the theory is carried out in the settingof Frechet spaces.
If the underlying manifold is a Lie group, one may restrict attention to deforma-tions compatible with the group structure. As it turns out, one obtains quantumgroups in the setting of C∗-algebras in this way [Rieffel 1993b; 1995]. We shallonly consider the case of compact Lie groups. A remarkable feature of the cor-responding compact quantum groups is that they arise from deformations of thealgebra of all smooth functions and not only of the algebra of representative func-tions. It is clear from the work of Rieffel that the deformed algebras of smoothfunctions fit naturally into the framework of bornological quantum groups.
Let G be a compact Lie group and let T be an n-dimensional torus in G withLie algebra t. We identify t with Rn and set V = Rn
× Rn . Let exp : t → T denotethe exponential map. Moreover let J be a skew-symmetric operator on V withrespect to the standard inner product. In order to obtain a Poisson bracket whichis compatible with the group structure of G we shall assume that the operator J isof the form J = K ⊕ (−K ) where K is a skew-symmetric operator on Rn
Using this data, the deformed product of f, g ∈ C∞(G) is defined by
( f ?K g)(x)=
∫f (exp(−K s)x exp(−K u)) g(exp(−t)x exp(v)) e2π i(〈s,t〉+〈u,v〉),
134 CHRISTIAN VOIGT
where the variables of integration range over Rn . This yields a continuous andassociative multiplication on C∞(G), and we write C∞(G)K for the correspondingbornological algebra. Together with the classical comultiplication, antipode, counitand Haar integral of C∞(G) this algebra becomes a bornological quantum group
We summarize this as follows and refer to [Rieffel 1993a, 1993b] for examples.
Proposition 11.1. Let G be a compact Lie group and let T be a torus in G withLie algebra t. For every skew-symmetric matrix K on t there exists a bornologicalquantum group C∞(G)K with structure as described above.
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[Rieffel 1995] M. A. Rieffel, “Non-compact quantum groups associated with abelian subgroups”,Comm. Math. Phys. 171:1 (1995), 181–201. MR 96g:46066 Zbl 0857.17014
[Voigt 2007] C. Voigt, “Equivariant periodic cyclic homology”, J. Inst. Math. Jussieu 6:4 (2007),689–763. MR 2337312 Zbl 05195938
[Woronowicz and Pusz 1999] S. L. Woronowicz and W. Pusz, “Analysis on the quantum plane: astep towards Schwartz space for the Eq (2) group”, notes from work in progress, 1999, Available athttp://www.fuw.edu.pl/~psoltan/prace/rgdr1999.pdf.
Received August 9, 2007. Revised October 4, 2007.
CHRISTIAN VOIGT
UNIVERSITAT MUNSTER
MATHEMATISCHES INSTITUT
EINSTEINSTRASSE 6248149 MUNSTER
GERMANY
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMALSURFACES
ADAM G. WEYHAUPT
The gyroid and Lidinoid are triply periodic minimal surfaces of genus threeembedded in R3 that contain no straight lines or planar symmetry curves.They are the unique embedded members of the associate families of theSchwarz P and H surfaces. We prove the existence of two 1-parameterfamilies of embedded triply periodic minimal surfaces of genus three thatcontain the gyroid and a single 1-parameter family that contains the Lidi-noid. We accomplish this by using the flat structures induced by the holo-morphic 1-forms G dh, (1/G) dh, and dh. An explicit parametrization ofthe gyroid using theta functions enables us to find a curve of solutions in atwo-dimensional moduli space of flat structures by means of an intermedi-ate value argument.
1. Introduction
The gyroid was discovered by Alan Schoen [1970], a NASA crystallographer in-terested in strong but light materials. Among its most curious properties was that,unlike other known surfaces at the time, the gyroid contains no straight lines or pla-nar symmetry curves [Karcher 1989; Große-Brauckmann and Wohlgemuth 1996].Soon after, Bill Meeks [1975] discovered a 5-parameter family of embedded genusthree triply periodic minimal surfaces.
Theorem 1.1 [Meeks 1975]. There is a real five-dimensional family V of periodichyperelliptic Riemann surfaces of genus three. These are the surfaces that canbe represented as two-sheeted covers of S2 branched over four pairs of antipodalpoints. There exist two distinct isometric minimal embeddings for each M3 ∈ V .
Meeks’s family contains many known examples of genus three triply periodicminimal surface, including the classical P, D, and CLP surfaces. Most memberswere previously undiscovered surfaces, and many have no straight lines and noplanar symmetries. Sven Lidin [1990] discovered a related surface, christened by
MSC2000: primary 53A10; secondary 49Q05, 30F30.Keywords: minimal surfaces, main/gyroid, lidinoid, triply periodic, flat structures.I am very grateful to my thesis adviser Matthias Weber for his patient help and encouragement.
137
138 ADAM G. WEYHAUPT
Lidin the HG surface but now commonly called the “Lidinoid”. Later, Große-Brauckmann and Wohlgemuth [1996] proved that the gyroid and Lidinoid areembedded.
Every currently known triply periodic minimal surface of genus three except forthe gyroid and Lidinoid is deformable, that is, for each triply periodic minimal sur-face M there is a continuous family of embedded triply periodic minimal surfacesMη for η ∈ (−ε, ε) such that M = M0, as long as M is neither the gyroid nor theLidinoid. In general, the lattices may vary with η, so that generically 3η1 6= 3η2 .We are primarily concerned with this question: do there exist continuous defor-mations of the gyroid and the Lidinoid? In general, the moduli space of genusthree triply periodic minimal surface is not understood, and the existence of thesedeformations would provide more information about the moduli space.
In a series of papers, the crystallographers and physical chemists Fogden, Hae-berlein, Hyde, Lidin, and Larsson graphically argue for the existence of two 1-parameter families of embedded triply periodic minimal surfaces that contain thegyroid and two additional families that contain the Lidinoid [Fogden et al. 1993;Fogden and Hyde 1999; Lidin and Larsson 1990]. While accompanied by veryconvincing computer-generated images, their work does not provide an existenceproof, and the mathematical landscape is fraught with examples where picturesmislead; see, for example, [Weber 1998].
The goal of this paper is to establish this main result:
Theorem 1.2. For η ∈ R+, there is a one parameter family of minimal embeddingsMη ⊂ R3/3η such that Mη is an embedded minimal surface of genus three. Thisfamily contains the gyroid. Each Mη admits a rotational symmetry of order 2.
We will call this family of surfaces the “tG” family following the notation of[Fogden et al. 1993]; the “t” stands for “tetragonal”. This theorem shows thatthe gyroid is deformable. Our other two main theorems prove the existence of aLidinoid family and an additional gyroid family.
Theorem 1.3. For η ∈ R+, there is a one parameter family of minimal embeddingsrLη ⊂ R3/3η such that rLη is an embedded minimal surface of genus three. Thisfamily contains the Lidinoid. Each rLη admits a rotational symmetry of order 3.
Theorem 1.4. For η ∈ R+, there is a one parameter family of minimal embeddingsrGη ⊂ R3/3η such that rGη is an embedded minimal surface of genus three. Thisfamily contains the gyroid. Each rGη admits a rotational symmetry of order 3.
As a consequence of these results, we have shown this:
All currently known examples of genus three triply periodic minimalsurfaces admit deformations.
None of these new examples are members of Meeks’s family.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 139
2. Preliminaries
2.1. Parametrizing minimal surfaces. For this section, refer to [Dierkes et al.1992; Osserman 1969; Nitsche 1975] for further details and history. Let � ⊂ C
denote a simply connected open domain, and let h = (h1, h2, h3) : � → C3 be anonconstant holomorphic map such that h2
1 +h22 +h2
3 ≡ 0 and |h1(z)|2 +|h2(z)|2 +
|h3(z)|2 6= 0 for all z ∈�. A direct computation shows that
(2-1) F :�→ R3, p 7→ Re∫ p
·
(h1dz, h2dz, h3dz)
is a minimal surface M ⊂ R3. The normal map N : M → S2 assigns to each pointp ∈ M the normal at p. The Gauss map, G : M → C ∪ ∞ is the stereographicprojection of the normal map.
To relate (2-1) to the geometry of the surface, we can rewrite the map as
(2-2) p 7→ Re∫ p
·
12
( 1G
− G, iG
+ iG, 1)
dh.
The meromorphic function G in (2-2) is the Gauss map
G = −h1 + ih2
h3.
Here dh is a holomorphic differential, often called the height differential. Givenany minimal surface M , there exists a height differential dh that, along with theGauss map, provides the above parametrization of a surface patch. Therefore,simply connected surface patches are fully parametrized.
The next result gives us a way to parametrize nonsimply connected surfaces.
Theorem 2.1 [Osserman 1969]. A complete regular minimal surface M havingfinite total curvature, that is, satisfying
∫M |K |d A <∞, is conformally equivalent
to a compact Riemann surface X that has finitely many punctures.
Since our triply periodic minimal surfaces M are compact in the quotient M/3,the fundamental domain necessarily has finite total curvature and therefore can beparameterized on a Riemann surface. Instead of using a simply connected domain� and meromorphic functions h1, h2, and h3, we instead consider three holomor-phic 1-forms ω1, ω2, and ω3 defined on a Riemann surface X , again with
∑ω2
i ≡ 0and
∑|ωi |
26= 0 (making sense of this first quantity pointwise and locally). We can
then define
F : X → R3, p 7→ Re∫ p
·
(ω1, ω2, ω3)
withω1 =
12
( 1G
− G)
dh, ω2 =i2
( 1G
+ G)
dh, w3 = dh.
140 ADAM G. WEYHAUPT
Since the domain is no longer simply connected, integration of the Weierstrassdata over a homotopically nontrivial loop γ on X is generically no longer zero. Thisintegration leads to a translational symmetry of the surface. The surface need not,at this point, be embedded or even immersed, so the term “symmetry” is perhapsmisleading here. More precisely, if F(p) = (q1, q2, q3) ∈ R3 for some choice ofpath of integration from the base point to p, then for any other choice of pathof integration, F(p) = (q1, q2, q3)+
∫γ(ω1, ω2, ω3) for some γ ∈ H1(X,Z). We
define the period of γ by
P(γ ) := Re∫γ
(ω1, ω2, ω3).
Fr a surface to be immersed and nonperiodic, we must have P(γ ) = 0 for allγ ∈ H1(X,Z). For a surface to be triply periodic with lattice 3 ⊂ R3, we musthave
P(γ ) ∈3 for all γ ∈ H1(X,Z).
Note
F1(z)+ i F2(z)= −
∫ z
·
G dh +
∫ z
·
1G
dh,
so the periods can be written as
P(γ )=
Re(−∫γ
G dh +∫γ
1G dh
)Im(−∫γ
G dh +∫γ
1G dh
)Re∫γ
dh
.The next lemma gives us a convenient way to generate surfaces.
Lemma 2.2. Let X be a Riemann surface of genus g. Let G : X → C ∪ ∞ bemeromorphic, and let dh be a holomorphic 1-form defined on X. Also assume thefollowing:
(1) If G has a zero or pole of order k at p, then dh also has a zero at p of orderk. Conversely, if dh has a zero of order k at p, then G must have a zero orpole of order k at p.
(2) There exists a lattice 3⊂ R3 such that P(γ ) ∈3 for all γ ∈ H1(X,Z).
Then the Weierstrass data (X,G, dh) define an immersed triply periodic minimalsurface of genus g.
The Weierstrass representation immediately allows for the following well-knownconstruction of a minimal surface. Let M0 be a minimal surface defined by Weier-strass data (X,G, dh). We construct a new minimal surface Mθ using Weierstrassdata (X,G, eiθdh). Note that the data still satisfies the requirements of Lemma2.2, in particular, that
∑j e2iθω2
j ≡ 0. The family of surfaces Mθ for 0 ≤ θ ≤ π/2
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 141
is called the associate family of M0. (Sometimes we say these surfaces Mθ are theBonnet transform of M0; see [Nitsche 1975; Bonnet 1853].) If the period problemis solved for M0, it will in general not be solved for Mθ , since PMθ
(γ ) is a linearcombination of PM0(γ ) and PMπ/2(γ ); a generic such linear combination need notbe either zero or in a lattice. The associate family plays an crucial role in theconstruction of the gyroid and Lidinoid.
2.2. Cone metrics. We call a flat structure with cone singularities a cone metric.When it is apparent from context that we are dealing with cone metrics, we willoften simply refer to a flat structure. Cone metrics are abundant for Riemann sur-faces — every holomorphic 1-form gives rise to a cone metric structure (recall thata Riemann surface of genus g has g linearly independent holomorphic 1-forms).
Proposition 2.3. Let X be a Riemann surface with meromorphic 1-form ω. LetUα be an open covering of X by simply connected sets, with distinguished pointspα ∈ Uα. Define gα : Uα → C by gα(z) =
∫ zpαω. Then (Uα, gα) endows X with
a cone metric (in fact, a translation structure). If ω has a zero or pole of order k,then this is a cone point of angle 2π(k + 1).
Proof. First, since Uα is simply connected, the integral∫ z
pαω does not depend on
the choice of pα — changing pα simply adds a constant.Away from ω’s zeros, gα is invertible, and so we have
gαβ(z)= z +
∫ pα
pβω = z + const,
which gives X a translation structure.The developing map of the flat structure is given by dev(γ ) =
∫γω. If ω has a
zero or pole at a point p (without loss of generality, p = 0), this developing mapextends meromorphically with pre-Schwarzian derivative
dev′′
dev′(z)=
dωω.
In the neighborhood of a zero or pole, we can locally write ω = zkh for a mero-morphic function h with h(0) 6= 0,∞. The residue of the pre-Schwarzian becomes
res0dev′′
dev′(z)= res0
kz
giving a cone of angle 2π(k + 1). �
2.3. Conformal quotients of triply periodic minimal surfaces. Our principal toolin the study of these surfaces will be taking the quotient of a triply periodic minimal
142 ADAM G. WEYHAUPT
surface by a rotational symmetry. We first recall Abel’s theorem, the Riemann–Hurwitz formula, and some corollaries. [Farkas and Kra 1992] contains a fulltreatment of the Riemann–Hurwitz formula.
Theorem 2.4 (Abel’s theorem). Let 0 be a lattice in C. There is an elliptic functionf on the torus C/0 with divisor
∑j n j Pj if and only if
(1)∑
j n j = 0, and
(2)∑
j n j Pj ∈ 0.
Theorem 2.5 (Riemann–Hurwitz formula). Let f : N ′→ N be a (nonconstant)
holomorphic map between a compact Riemann surface N ′ of genus g and a com-pact Riemann surface N of genus γ . Let the degree of f be n. Define the totalbranching number of the mapping to be B =
∑P∈N ′ b f (P). Then
g = n(γ − 1)+ 1 + B/2.
Corollary 2.6 [Farkas and Kra 1992, V.1.5]. For 1 6= T ∈ Aut(M),
|fix(T )| ≤ 2 +2g
order(T )− 1+
2γ order(T )order(T )− 1
with equality if order(T ) is prime.
Corollary 2.7 [Farkas and Kra 1992, proof of V.1.5].
(2g − 2)= order(T )(2γ − 2)+order(T )−1∑
j=1
∣∣fix(T j )∣∣.
Using these tools, we can describe the quotient of a minimal surface by a rota-tional symmetry.
Proposition 2.8. Let M be an embedded triply periodic minimal surface admit-ting a rotational symmetry ρ with axis of symmetry x3. Then the quotient surfaceM/3/ρ has genus one.
Proof. We use the Riemann–Hurwitz formula. Notationally, N ′ is M/3, N isM/3/ρ, and f is the quotient map f : M/3→ M/3/ρ. Note that γ 6= 3, sincef is not degree 1. Similarly, if γ = 2, then by Riemann–Hurwitz, 2 = n + B/2,and so either n = 2 and the map is unbranched or n = 1 (which is impossible sincerotational symmetries have order at least 2). If n =2, then by Corollary 2.6 the mapρ must have 4 fixed points on M/3, implying that f is branched, a contradiction.
Furthermore, the quotient cannot have genus γ = 0, since the height differen-tial dh is invariant under ρ; therefore it descends holomorphically to the quotientM/3/ρ. Of course, a surface of genus 0 has no holomorphic differentials, soγ 6= 0. The only remaining possibility is γ = 1. �
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 143
We can use these results to obtain information about the Gauss map of a minimalsurface. Let M/3 be an embedded genus three triply periodic minimal surface,and let ρ ∈ Aut(M) with order(ρ)= 2. Using a rigid motion, we orient M so thatthe axis of rotational symmetry is the x3-axis. By Proposition 2.8, M/ρ is a torus,and so, by Corollary 2.6, ρ has exactly 4 fixed points. The fixed points are preciselythose points with vertical normal, and we scale M so that the torus generators are1 and τ with τ ∈ C ∩ {Im τ > 0}. The squared Gauss map G2 descends to thequotient torus.
Lemma 2.9. G2 has two first-order poles and two first-order zeros.
Proof. Part (1) of Abel’s theorem tells us that there must be an equal number ofzeros and poles. Suppose, by way of contradiction, that G2 had a second-orderzero at 0. Thus G has a zero of at least second order on the genus three surfaceM . Since dh is the lift of dz and since dz has no zeros, dh has zeros of at mostfirst order on the genus three surface in space (locally, the pullback map behaveslike z2 at a branch point). However, for the metric on M to be nondegenerate andto have no ends, dh must have a zero of at least second order, a contradiction.Therefore, the zeros of G2 are of at most first order. The same reasoning holds forthe first-order poles. �
3. Parametrization of the gyroid and description of the periods
The gyroid is the unique embedded member in the associate family of the SchwarzP surface (except for the D surface, the surface adjoint to the P surface). Therefore,to parametrize the gyroid, we must first understand the P surface. We now give aparametrization of the Schwarz P surface by a convenient Weierstrass representa-tion. Then we describe the gyroid as a specific member of the associate family ofthe P surface. Finally, we describe the periods of the gyroid in terms of the flatstructures.
3.1. The P Surface and tP deformation. The Schwarz P surface (see Figure 3.1)can be constructed in a number of different ways. The approach taken below, whileuseful for our purposes, is not the most direct parametrization.
The P surface admits an order 2 rotational symmetry ρ2 : R3→ R3 about the
x3-axis. Since the rotation is compatible with the action of 3 on R3, ρ2 descendsto an order 2 symmetry of the quotient surface P/3 (abusing notation, we alsocall the symmetry induced on the quotient ρ2). ρ2 has four fixed points on P/3as illustrated in Figure 3.2. (The fixed points of a rotation about a vertical axisare exactly those points with vertical normal. For any genus three triply periodicminimal surface, there are at most four points with vertical normal since the degreeof the Gauss map is 2.) The quotient P/3/ρ2 is a (conformal) torus C/0 (compare
144 ADAM G. WEYHAUPT
Figure 3.1. Left: A translational fundamental domain of theSchwarz P surface. Right: A fundamental domain of the SchwarzH surface.
A
A
A
1 1
B
B
B
2
2
3
3
Figure 3.2. P surface with generators for the homology. Fixedpoints of ρ2 are shown in red.
Proposition 2.8 and Corollary 2.6, noticing that ρ2 is not the hyperelliptic involutionsince it fixes only four points).
The lattice 3 is the cubical lattice generated by the unit length standard basisvectors {e1, e2, e3}. We can restrict the possible conformal structure of the torusC/0 by considering reflectional symmetries. The P surface admits a reflectionalsymmetry that also commutes with ρ2, namely, the reflection in the plane contain-ing x1 and x3. Its fixed point set consists of two disjoint totally geodesic curves.Since this reflection commutes with ρ2, it descends to the torus C/0 as a symmetry,which yields two disjoint fixed point sets. The only conformal tori that admit two
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 145
disjoint fixed point sets of a single (orientation reversing) isometry are the rectangu-lar tori (rhombic tori admit orientation reversing isometries with a connected fixedpoint set). Therefore, 0 is generated by b ∈ R and τ ∈ i · R. Since the conformalstructure is unchanged by a dilation in space, we may dilate so that we can take0 = 〈1, τ 〉 with τ = ai for a ∈ R. (Note that the dilation required to normalize thetorus this way may change the lattice 3 so that the generators no longer have unitlength.) The map P/3→ (P/3)/ρ2
= C/0 is a branched covering map. We canidentify (using the aforementioned symmetries) the location (on the torus) of thebranch points of this map: branch points corresponding to zeros of G are locatedat 0 and τ/2, while branch points corresponding to poles of G are located at 1/2and 1/2 + τ/2.
Since the x3 coordinate is invariant under ρ2, the height differential dh descendsholomorphically to the quotient torus as reiθdz for some r ∈ R and 0 ≤ θ ≤ π/2(since dz is, up to a constant multiple, the only holomorphic 1-form on C/0).Varying r only scales the surface in space, and so r is determined by our require-ment that one of the generators of the torus is 1 (we will drop the r for the restof this work, since scaling is inconsequential to us). θ is the important Bonnettransformation parameter. For the P surface, θ = 0. As noted in Lemma 2.9, thesquared Gauss map G2 has simple poles and zeros at the branch points.
We can explicitly write the formula for G2 using theta functions as
G2(z) := ρθ(z, ai) θ(z − (a/2)i, ai)
θ(z − 1/2, ai) θ(z − (1 + ai)/2, ai).
The multivalued function G on C/0 is obtained by G(z) =
√G2(z). The factor
ρ is called the Lopez–Ros factor and gives rise to many interesting deformationsof minimal surfaces, most of which are not embedded [Lopez and Ros 1991]. Ifρ = r1eiφ , varying φ simply produces a rotation of the minimal surface in space.We will use φ indirectly to normalize certain quantities. We will also determinethe real part r of ρ by a normalization, although varying r is highly destructive:in general, if a surface is embedded for ρ = ρ0, modifying ρ will instantly yielda nonimmersed surface (a la the Bonnet transformation). We will determine anappropriate value of ρ for the P surface in Section 3.1.3.
The torus and the branch points are invariant under the symmetry −id; thequotient S = (C/0)/(−id) is a sphere with 4 branch points.
The 1-forms G dh, (1/G)dh, and dh each place a flat structure on the toruswhich, after taking the quotient with −id, descends to the sphere. We study herethe developed image of each flat structure, which we will then use to computeperiods. We study each flat structure independently.
3.1.1. dh flat structure for the P surface. Since the dh flat structure descends aseiθdz, the developed image of the flat structure for the torus is simply the rectangle.
146 ADAM G. WEYHAUPT
Consider the “lower half” of the rectangle as a fundamental domain for the action−id , and note the additional identification induced. One can then see directly thesphere S. The dh flat structure is, in fact, a rectangle with τ/2 directly above0. This is because there is a horizontal plane of reflection that interchanges thezeros and poles of the Gauss map. The horizontal symmetry curve descends tothe quotient torus as a vertical straight line. The reflection only interchanges thebranch points if the torus is oriented so that the points corresponding to 0 and τ/2in the developed flat structure have the same imaginary part.
3.1.2. G dh flat structure for the P surface. As noted in the proof of Proposition2.3, the order of the zeros and poles of the 1-form G dh produce cone angles on thetorus of 3π at both 0 and τ/2 and of π at both 1/2 and 1/2 + τ/2. The involution−id halves the cone angles in the quotient, so that on the sphere the cone anglesare
• a cone point of angle 3π/2 at both 0 and τ/2;
• a cone point of angle π/2 at both 1/2 and 1/2 + τ/2.
The situation is illustrated in Figure 3.3.Developing the sphere with cone metric induced by G dh gives a hexagon:
τ2
0
12
1+τ2
Figure 3.3. A three-dimensional topological picture of the sphereS with flat structure induced by G dh. Note that the angles are notdrawn correctly. Cone points are visible at the marked vertices.Thick black lines indicate cuts made to develop the sphere (tetra-hedron) into the plane. Although this is conformally not the G dhcone metric, this tetrahedron is conformally the development ofthe dh cone metric (since all points are regular for dh).
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 147
Lemma 3.1. By cutting along the shortest geodesics on the sphere from 1/2 to 0,0 to τ/2, and from τ/2 to 1/2 + τ/2 and developing into the plane, we obtain ahexagon shown in Figure 3.4. The hexagon has the properties
(i) the length of li is equal to that of l∗i for i = 1, 2, 3;
(ii) the angle between l1 and l2 and the angle between l∗1 and l∗2 are both 3π/4;
(iii) the angle between l1 and l∗1 and the angle between l3 and l∗3 are both π/2.
The proof is nearly identical to the more general proof of Lemma 4.1 and isomitted.
At this point, we have not yet determined the value of a for the torus. Whatis clear is that once a is chosen, the entire G dh flat structure will be fixed. Forthe moment, we describe the flat structures and study the period problem with thisdeterminacy still unresolved.
3.1.3. (1/G)dh flat structure for the P surface. G2(z + 1/2) and (1/G)2(z) haveprecisely the same zeros and poles to the same order. By Liouville’s theorem, thequotient is constant, since it is holomorphic (with no poles) and doubly periodic.Thus
G2(
z +12
)/( 1G
)2(z)= r1eiφ1 .
By adjusting the Lopez–Ros parameter, we can ensure that this factor is 1, andwe do that for the P surface. Therefore, the (1/G)dh flat structure is simply atranslation of the (infinite, periodic) G dh flat structure. This is reflected in theoutline of the (1/G)dh flat structure in Figure 3.4.
3.1.4. Compatibility of G dh, (1/G)dh, and dh. We have drawn G dh and dhoriented a specific way; namely, the dh flat structure is horizontal, and the G dhflat structure has the line segment l2 vertical. We have not yet justified the secondof these claims. More generally, any time one prescribes all three data — G dh,(1/G)dh, and dh — one has to ensure that G dh · (1/G)dh = dh2. This compati-bility is a serious problem when showing the existence of surfaces in general, butthe approach taken in Section 4.2 avoids this problem completely.
For the P surface, one can see that this orientation is correct as follows. There isa vertical symmetry plane that interchanges the two zeros of the Gauss map. Thisreflection descends to the torus, and the symmetry curve is exactly the horizontalline at y = Im τ/4 (recall we have fixed a fundamental domain of the torus). Aftera translation, dh is real on this symmetry curve. Under the flat structure G dh, thissymmetry curve develops to the line segment from (0, q/2) to (2p, q/2). Again,after a vertical translation,
∫G dh is real on this segment. This is also true for
(1/G)dh (the developed flat structure is only a translation of that for G dh). Thuswe see that both G dh and (1/G)dh are real on this segment, and this is compatible
148 ADAM G. WEYHAUPT
(p,−p)
(0, 0)
(0, q)
(p, p + q)
(2p, q)
(2p, 0)
l2 l∗2
l1 l∗1
l3 l∗3
Figure 3.4. The G dh (thick lines) and (1/G)dh (thin lines) flatstructures for the P surface. Labeled vertices are for the G dh flatstructure; the corresponding points on (1/G)dh are obtained bytranslation by (−p, p).
with dh. (The only possible inconsistency is the rotational orientation of G dh, soit suffices to check one curve.)
3.2. The period problem for the P surface. The six cycles shown in Figure 3.2generate the homology H1(P/3,Z). Figure 3.5 shows these cycles on the 2-sheeted branched torus, along with cuts to identify this structure with the surfacein space. To compute the periods, we need to compute
∫γ
G dh for each generatorγ of the homology (and do the same for (1/G)dh). Since
∫G dh is simply the
developing map of the G dh flat structure on the torus, we can compute in termsof the cycles’ image on the developed flat structure. To calculate the periods, wefirst obtain the horizontal contribution from the G dh and (1/G)dh flat structures,for example∫
A1
G dh = (1+ i)(p + pi)= 2p · i and∫
A1
1G
dh = (1− i)(p − pi)= −2p · i.
The vertical periods are easily read off of the torus as simply the difference inthe endpoints of the curves drawn on the torus in Figure 3.2. Recalling that
P(γ )=
(Re(−∫γ
G dh +∫γ
1/Gdh), Im
(−∫γ
G dh +∫γ
1/Gdh),Re
∫γ
dh),
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 149
A1
A2
A3
B3B1
Sheet 1
B2
Sheet 2
Figure 3.5. The conformal model of the P surface showing thehomology generators. Cuts to reconstruct the surface by gluing areshown by dashed lines. Branch points corresponding to zeros ofthe Gauss map are shown by solid dots, while poles are indicatedby an X.
we write
P(A1)= (0, 0, 0), P(B1)= (2(p + q),−2(p + q), 0),
P(A2)= (0, 0, 0), P(B2)= (−2(p + q),−2(p + q), 0),
P(A3)= (0, 0, 1), P(B3)= (0, 0, 0).
(This last horizontal period is zero due to the 2-fold symmetry of this curve. Sincethe cycle continues onto both sheets, we develop from p1 to p4, then rotate 180◦
(to get on the other sheet), then develop the same length again. This causes thehorizontal period for B3 to vanish.) It is immediately clear that these periods gen-erate a 3-dimensional lattice 3 for all values of p and q. In other words, theperiod problem is solved no matter what the actual lengths of the segments in thedeveloped flat structure are. Thus any value of a (and therefore, any quotient torus)solves the period problem.
150 ADAM G. WEYHAUPT
While we have phrased this section as if we were describing the P surface, whatwe have actually seen is that there is a family of immersed triply periodic minimalsurfaces that contains the P surface.
Theorem 3.2. There exists a continuous family of embedded triply periodic min-imal surfaces of genus three that contains the P surface (the tP family). Eachmember of the family admits an order 2 rotational symmetry and has a horizontalreflective symmetry plane.
This is a consequence of the fact that all rectangular tori (with the given Weier-strass data) solve the period problem. (Embeddedness follows from Proposition4.6.) Note that all of these surfaces are in the Meeks family.
We will call this family of minimal surfaces the tP family (Figure 3.6). Notethat the limit τ → 0 looks like a pair of parallel planes joined with small catenoidalnecks. The limit τ → ∞ looks like a pair of perpendicular planes that are desin-gularized along the intersection by adding handles (like the singly periodic Scherksurface). See [Traizet 2008] for interesting results related to these limiting surfaces.
3.3. The gyroid. We are finally ready to describe the gyroid minimal surface.Schoen [1970] describes a surface that is associate to the P and D surfaces andis embedded. Let (X,G, dh) be the Weierstrass data describing the P surface(see Section 3.1). Recall that a surface is called associate to (X,G, dh) if itsWeierstrass data is (X,G, eiθdh). For a single value of θ , this associate surface isan embedded minimal surface, which Schoen called the gyroid. In his descriptionof the gyroid, Schoen estimated θ ≈ 38.0147740◦. That this value of θ along withθ = 0 (P surface) and θ = π/2 (D surface) are the only values that produce anembedded minimal surface is something of a curiosity. In [Große-Brauckmannand Wohlgemuth 1996], the gyroid is described geometrically as follows. Theangle of association for the gyroid is such that the vertical period of B3 must betwice that of A3. Since the B3 curve continues on both sheets of the torus, weneed the images of the curves on a single sheet of the developed image of the dhflat structure of the torus to have equal real part. This is equivalent to choosing θso that the rotated dh flat structure places the point 1 + τ directly above 0 in thedeveloped image (see Figure 3.7). Therefore θ = arccot Im τ .
Unfortunately the value of τ that gives the standard, most symmetric P surfacestill must be determined by an elliptic integral. In other words, if (X,G, dh) is anymember of the tP family, then (X,G, eiθdh) is an embedded surface only if θ = 0,θ = π/2, or (X,G, dh) describe the most symmetric (what we call the “standard”)P surface. As an unfortunate consequence of this fact, we see that varying τ ∈ i ·R
is not enough to yield a family of gyroids — we must consider τ ∈ C.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 151
τ = 0.24 τ = 0.40
τ = 0.781 τ = 1.10
τ = 1.70 τ = 3.00
Figure 3.6. The tP deformation of the P surface, for different val-ues of τ .
A quick computation using the resulting flat structures gives the following pe-riods for the gyroid:
(3-1)
P(A1)= (1, 0, 0),
P(A2)= (1, 0, 0),
P(A3)= (0, 1, 1),
P(B1)= (1, 0,−1),
P(B2)= (−1, 0,−1),
P(B3)= (0, 0, 2).
152 ADAM G. WEYHAUPT
eiθ · 1eiθ · τ
Figure 3.7. The alignment of the dh flat structure for the gyroid.The alignment of the dh flat structure for the gyroid.
We have gone through a fairly complicated set of gymnastics to show that thegyroid is even immersed, but this complicated method does nicely set up the periodproblem for Section 4.2, where we prove the existence of two families of gyroids.There is, however, a much easier way to see that the gyroid is immersed. Insteadof considering the flat structures induced by G dh, (1/G)dh, and dh, considerinstead the Weierstrass 1-forms ω1, ω2, and ω3 = dh. The standard P surfaceadmits an order 3 rotational symmetry that interchanges each of the coordinateaxes; the action of this rotation on the space of holomorphic forms permutes the1-forms ωi . The flat structures, therefore, are all congruent, and all of the periodscan be expressed in terms of these 1-forms. Since the associate family parameter θsolves the vertical period problem and since the flat structures of these forms are allcongruent, the period problem is completely solved. Even though this brief proofeasily shows that the gyroid is immersed, this technique seems to fail miserably atachieving a family of gyroids. As soon as we lose the symmetry of the standard Psurface, the technique is no longer useful.
4. Proof of main theorem
In this section, we prove Theorem 1.2, which says there exists a family of gyroidsthat preserve an order 2 rotational symmetry.
4.1. Sketch of the proof. First, define a moduli space of polygons H(G) that solvethe horizontal period problem. That is, suppose X is a Riemann surface constructedas the branched (double) cover of a torus T , a Gauss map G, and a height differen-tial dh so that the developed image of the torus T with cone metric induced by G dhis in H(G). Then the horizontal periods will lie in a lattice. Also, the generatorsof the lattice will be the periods of the same cycles that generate the lattice for thegyroid. (In fact, our tori will have the property that they are invariant under −id,
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 153
so we only develop T/−id .) Note that the lattice will not be constant throughoutthe deformation. Since the horizontal period problem requires knowledge aboutnot only G dh but also (1/G)dh, we impose a normalization so that the (1/G)dhflat structure is a translate of the flat structure developed by G dh.
Second, define a moduli space of polygons V(G) that solve the vertical periodproblem. Since the conformal model of a Riemann surface is, in our case, alwaysthe (two or three)-fold cover of a torus, the vertical moduli space will always consistof parallelograms. The critical issue here will be the orientation of the developedimage of the parallelograms. (In fact, orientation of the developed flat structure isalso the critical issue for H(G).)
Then, we show that there exists a set of Weierstrass data {Xη,Gη, dhη} for η∈ R
such that the developed image of the torus Tη under the flat structure inducedby G dh is in H(G) and that induced by dh is in V(G). This shows that boththe horizontal and the vertical period problems can be solved simultaneously bya family of Weierstrass data. To accomplish this, define a continuous functionh : C → R with the property that h(τ ) = 0 implies that there exists θ(τ ) suchthat the G dh flat structure (respectively the dh flat structure ) will be in H(G)(respectively V(G)) provided that dh = eiθ . The Gauss map will be determined bythe conformal structure of the torus and the normalization requiring that the G dhand (1/G)dh flat structures are translates. We then show that h−1(0) contains acurve and that this curve contains the value τ that determines the standard gyroid.This guarantees the existence of a continuous family of immersed triply periodicminimal surfaces that contains the gyroid. To study the zero set of h, we usethat we can compute h more or less explicitly for rectangular tori. Also, we cancompute h for τ = n + yi for n ∈ Z by studying the effect of twists on the torusand the flat structures. This allows us to compute sufficiently many values to usean intermediate value argument.
Finally, we show that the surfaces obtained in this way are embedded (and notjust immersed), a consequence of the maximum principal for minimal surfaces; seethe survey [Lopez and Martın 1999]. We separate the embeddedness portion of theproof into the more general Proposition 4.6.
In Section 4.2, we set up the moduli spaces H(G) and V(G). In Section 4.3,we prove the remaining statements. We do this in detail for the tG family. For thefamilies rG and rL, we construct the moduli spaces in Section 5.
4.2. Horizontal and vertical moduli spaces for the tG family. In the most generalsetting, it is not possible to split the period problem into vertical and horizontalcomponents. In our case however, we are considering surfaces that are invariant un-der rotation. Therefore, since the height is invariant under this rotation, the heightdifferential dh establishes a consistent x3 direction that is invariant throughout the
154 ADAM G. WEYHAUPT
family. Therefore, the lattice 3 is a product Z×31, and so we can split the periodproblem into two parts. We need to show that there is a single vertical period, andthat the horizontal periods lie in a two-dimensional Z-lattice.
4.2.1. Definition of V(G) and calculation of the vertical periods. In this subsec-tion we describe the conformal models of the surfaces we wish to construct. Recallthat the underlying Riemann surface structure for the gyroid was a 2-fold branchedcover of a rectangular torus that parametrized the P surface (here we are referringto the most symmetric P surface).
Denote by V(G) the space of marked parallelograms in C up to equivalence bytranslations (we consider marked parallelograms to distinguish the cone point 0).Notice that if 0 = 〈1, τ 〉 is a Z-lattice in C, the torus C/0, once equipped withthe flat structure induced by eiθdz, develops to an element of T ∈ V(G). If M is atriply periodic minimal surface with symmetry ρ2 such that M/3/ρ2 = C/0, andif we develop generators of the homology H1(M,Z) onto T , then M is immersedonly if both the horizontal and the vertical period problems are solved. The periodproblem is in general not solved if C/0 develops into V(G).
We will now define a subset of V(G) that does solve the vertical period problem.There are generally many such subsets, but we seek a deformation of the gyroid.Recall (see Figure 3.7) that the gyroid’s dh flat structure for the torus satisfiesRe eiθ
= − Re eiθτ . With this motivation, we define
V(G)= {(ω1, ω2) ∈ C × C | |ω1| = 1 and Reω1 = − Reω2}.
Developing the cycles shown in Figure 3.2 onto this flat structure, one easilysees that the vertical period problem is solved. Using the notation of the cyclesfrom Figure 3.5, the vertical periods are
P(A1)= (−−,−−, 0),
P(A2)= (−−,−−, 0),
P(A3)= (−−,−−,Reω1),
P(B1)= (−−,−−,− Reω1),
P(B2)= (−−,−−,− Reω1),
P(B3)= (−−,−−, 2 Reω1).
4.2.2. Definition of H(G) and calculation of the horizontal periods. Suppose thatM is any immersed, genus three, triply periodic minimal surface that has as aconformal model a two-fold branched cover of a generic torus C/0. Without lossof generality we write 0 = 〈1, τ 〉. Suppose further that the square of the Gaussmap descends to C/0 and has simple poles at 1/2 and 1/2+τ/2 and simple zerosat 0 and τ/2. (This is the case for the gyroid, except that the torus is rectangular.)The quotient S = C/0/(−id) is a sphere, and G dh again induces a cone metric onS. Under this cone metric, the sphere is a tetrahedron with two vertex angles 3π/2and two vertex angles π/2. The developed image of this sphere has a particularlynice parametrization:
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 155
ξ1
(0, 0)
ξ2
l1
l∗1
ξ1
l2
l∗2
l3
l∗3
α
α∗
Figure 4.1. A generic member of the H(G) moduli space.
Lemma 4.1. For any torus C/0 with 0 = 〈1, τ 〉 the cone metric G dh descends toS. By cutting along shortest geodesics on S from 1/2 to 0, 0 to τ/2, and from τ/2to 1/2+τ/2, we obtain a hexagon; see Figure 4.1. The hexagon has the properties
(i) the length of li is equal to that of l∗i for i = 1, 2, 3;
(ii) l2 is parallel to l∗2 ;
(iii) the angle between l1 and l∗1 and the angle between l3 and l∗3 are both π/2.
We can parametrize the space of possible hexagons by ξ1, ξ2 ∈ C as shown inFigure 4.1.
We call the space of all hexagons satisfying the conditions of Lemma 4.1 H(G).
Proof. Since C/0/(−id) is a sphere and since −id fixes the branch points of C/0,the flat structure induced by G dh makes S a tetrahedron with cone angle 3π/2 at0 and τ/2 and with cone angle π/2 at 1/2 and (1+τ/2). By making the indicatedcuts, we obtain a hexagon with sides l1, l2, l3, l∗1 , l∗2 , and l∗3 . We denote the points inthe developed image corresponding to 1/2, 0, τ/2, and (1+τ/2)∈ C/0 by p1, p2,p3, and p4, respectively. By making a translation, we arrange so that p2 = 0 ∈ C.Each li was identified with l∗i before the cutting; therefore, the length of li is equalto l∗I . Also, since there is a π/2 cone angle at 1/2, the angle between lines l1 andl∗1 must be π/2 (and similarly for l3).
Let α denote the angle between l1 and l2, and let α∗ denote the angle between l∗1and l∗2 . The cone angle at 0 is 3π/2, therefore, since both p2 and p∗
2 correspond to
156 ADAM G. WEYHAUPT
0 ∈ T , the sum α+α∗= 3π/2. One can see this by developing a small circle about
0 and noting that in the developed image we must obtain an arc that subtends anangle of 3π/2. �
To understand the horizontal periods, we again adjust ρ, if necessary, to normal-ize the (1/G)dh flat structure as in Section 3.1.3 so that the developed flat structurefor (1/G)dh is simply a translate of that for G dh; ρ is uniquely determined bythis normalization. Then in terms of these flat structures, we compute the periodsof the six generators of H1(M, Z) and tabulate them:
X∫
X G dh∫
X (1/G)dh
A1 (1 + i)(ξ1 − ξ2) (1 − i)(ξ2 + iw− ξ1 + ξ1)
B1 (1 + i)(ξ1 − ξ1) (1 − i)ξ2
A2 (i − 1)ξ1 (1 − i)ξ1
B2 (1 − i)ξ2 (1 + i)(ξ1 − ξ1)
A3 (−1 − i)ξ1 (−1 − i)ξ1
B3 2(ξ1 − ξ1) 2ξ2
The notation ξ1 is the complex number corresponding to p4 (see Lemma 4.1),that is,
ξ1 = − ξ1 + ξ2 +(2 + 2i)ξ 2
1 ξ1
2|ξ1|2.
To simplify the calculations, we make the change of variables
a = 2(Re ξ1 + Im ξ2) and b = 2(Im ξ1 − Im ξ2).
One can then compute the horizontal periods to be
PA1 = (a + b, 0,−−),
PB1 = (a, b,−−),
PA2 = (a + b, 0,−−),
PB2 = (−a, b,−−),
PA3 = (0, a + b,−−),
PB3 = (0, 0,−−).
Notice that when b = 0, the period problem is solved. In particular, when b = 0the periods coincide with those of the gyroid; see Equation (3-1)) Recall that b =
2(Im ξ1 − Im ξ2); define
H(G)= {(ξ1, ξ2) ∈ H(G) | Im ξ1 = Im ξ2}.
Then every flat structure in H(G) solves the horizontal period condition, and doesso with the same relations among the generating curves as for the gyroid. Figure4.2 shows a typical member of H(G).
Of course, there are other choices for a and b that also solve the horizontalperiod problem. We make this choice because we want the family to contain thegyroid. The choice a = −b, for example, would yield the tP family of Section 3.1.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 157
(0, 0)
ξ1ξ2
Figure 4.2. A generic member of the H(G) moduli space.
We have shown that if (X,G, dh) has as flat structures members of H(G) andV(G), then the period problem is solved. Certainly the Weierstrass data for thegyroid do solve the period problem. It remains to find a 1-parameter family ofsuch data. We will then show that the surfaces are all embedded.
4.3. Proof of the tG family. To prove the existence of the tG family, our first taskis to show that there exists a family of Weierstrass data (X,G, dh) with X thedouble branched cover of a torus such that the developed image of C/0/(−id)under G dh is in H(G) and such that the developed image of C/0/(−id) underdh is in V(G). This will show that the period problem is solved.
Let 0 = 〈1, τ 〉. Define dh = dz on C/0. Define Xτ to be the Riemann surfaceobtained from the double cover of C/0, with branch points at 0, 1/2, τ/2, and1/2 + τ/2 and with branch cuts as shown in Figure 3.5. The square of the Gaussmap will be well defined on C/0 as the unique meromorphic function with zerosat 0 and τ/2 and poles at 1/2 and 1/2 + τ/2, up to a complex multiple ρ. Defineρ so that the G dh and (1/G)dh flat structures are normalized as in Section 3.1.3,that is, so that they are translates.
For each choice τ ∈ C, this data describes a minimal surface.
Definition 4.2. The vertical relative turning angle θV(τ ) is
θV(τ ) := π/2 − arg(1 + τ).
This is precisely the angle by which the dh flat structure fails to be in V(G).The horizontal relative turning angle θH(τ ) is the angle by which the G dh flat
structure must be rotated so that it satisfies Im ξ1 = Im ξ2.
If θV(τ ) = θH(τ ), then we could define dh = eiθV(τ )dz = eiθH(τ )dz. The defi-nition of horizontal and vertical turning angle ensures that (X,G, dh) solves thehorizontal and vertical period problem. Define b(τ ) := θH(τ )− θV(τ ).The periodproblem is solved exactly on the zero set of b. Let τG denote the value of τ whichyields the gyroid; τG ≈ 0.781i .
158 ADAM G. WEYHAUPT
B
Y
L R
gyroid
//
Figure 4.3. b < 0 on Y and b > 0 on B, so the zero set containsa curve separating B and Y (it must pass through the value thatyields the gyroid).
Our goal is to understand the zero set of b. Note that when τ ∈ iR, the resultingtorus is rectangular. On rectangular tori, it is possible to explicitly develop the conemetric G dh into C by integrating the Gauss map (recall that the Gauss map canbe explicitly given in terms of theta functions) and to therefore understand b. On ageneric, nonrectangular torus explicit computation is not possible, since the edgesof a fundamental domain are no longer fixed point sets of an isometry and thusare not totally geodesic. It is no longer the case that these edges of a fundamentaldomain develop, under integration, to the shortest geodesic between cone pointsof the tetrahedron.
Next, as in Figure 4.3, consider the half plane, with the y-axis divided into twosegments B and Y , where Y = {(0, y) | Im(y) < Im τG}.
Lemma 4.3. b > 0 on L and R, where L is a vertical line x = −1 and R is avertical line x = n for n ∈ Z sufficiently large.
Proof. The quotient sphere of the torus generated by (1, 1 + τ) is related to thesphere obtained from the torus generated by (1, τ ) by performing a Dehn twist onthe cycle A1. To understand the effect of the Dehn twist on the G dh flat structure,note that after the twist, ξ2, ξ1 + ξ1, and ξ1 are translated by ξ1 + ξ1 − ξ2, while theremaining vertices of the developed flat structure are fixed.
We can compute the G dh flat structure explicitly in the case of normalizedrectangular tori and flat structures: for all rectangular tori, the angle between l1
and l2 is 3π/4 and when normalized (recall that this requires that the G dh and(1/G)dh flat structures are aligned), the segment l2 is vertical with Im ξ2 > 0.This last is a consequence of the symmetries of rectangular tori; see Section 3.1.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 159
l1 l∗1
l∗2
l∗3l3
l2 0
ξ1
ξ2
Figure 4.4. Left: The G dh flat structure after many positive Dehntwists. Right: A “rectangular torus” G dh flat structure after ap-plying a single negative Dehn twist.
After a large number of positively oriented Dehn twists, we see a G dh flatstructure as in Figure 4.4. Therefore θH(n + τ)≈ π for large n ∈ Z. This value islarger than θV(n + τ)≈ π/2. Thus b ≈ π/2> 0 on R.
Showing that b> 0 on L is similar other case. After applying a single negativelyoriented Dehn twist, θH remains positive (θH is always positive), but θV(−1+ci)=0 for all c ∈ iR. Therefore b> 0 on L , since a negatively oriented Dehn twist shiftsthe “top cone” of the G dh flat structure; see Figure 4.4. �
Lemma 4.4. b > 0 on B and Y .
Proof. We will show that both θV and θH are monotone — decreasing and increas-ing, respectively — as Im τ increases. This implies that the b has at most 1 zero.Of course, we know that a zero occurs at τG , yielding the gyroid.
Fortunately, we are able to explicitly calculate θV as θV(τ )= arccot(Im τ). Thisfunction is clearly monotone decreasing in Im τ .
The situation for the horizontal turning angle is not as simple. First, for allτ ∈ iR, the G dh flat structure is normalized in the same orientation as it is for thegyroid, that is, the straight line segment λ from the developed image of 0 to thedeveloped image of 1 is horizontal. To see this, observe that in the rectangularcase, there is a vertical symmetry curve in space that translates, on the torus,to a horizontal curve (straight line) connecting 0 and 1. The endpoints have nohorizontal displacement, and so must be conjugate in the G dh and (1/G)dh flatstructures. Since all surfaces in the tP family share this symmetry, the line λ mustbe horizontal.
Since for all τ ∈ iR the normalized G dh flat structure is aligned so that λis horizontal when the angle of association is 0, the relative turning angle in therectangular case is computed in terms of the ratio of |l1| to |l2|; precisely,
θH(τ )= π − arg((|l2|/|l1|)i − e−iπ/4).
160 ADAM G. WEYHAUPT
f1 g1
p1 p2 1
τ1
f2 g2
p1 p2
p3
1
τ2
Figure 4.5. Two maps from the upper half plane to a hexagon.
Therefore θH(τ ) increases as the ratio |l2|/|l1| increases. We now show that |l2|/|l1|
is monotone in Im τ .Suppose that there exist τ1, τ2 ∈ iR such that |lτ1
2 |/|lτ11 | = |lτ2
2 |/|lτ21 |. Because of
the restrictions on the flat structures imposed by the rectangular torus (see Lemma3.1), this implies that the G dh flat structures are dilations of each other. Callthe developing map from the torus C/〈1, τi 〉 to the plane (yielding a hexagon) gi .The Schwarz–Christoffel maps fi map the upper half plane to the tori C/〈1, τi 〉.Composing gives two maps from the upper half plane to similar hexagons; seeFigure 4.5.
Thus g1 ◦ f1 and 12 · g2 ◦ f2 are two maps from the upper half plane to the
same hexagon. By the Riemann mapping theorem, there is a unique such mapup to Mobius transformation, which is however fixed because both maps send p1
to the same point. Since g1 ◦ f1(p2) =12 · g2 ◦ f2(p3) these maps are distinct (a
contradiction with the Riemann mapping theorem) unless p2 = p3. But these pointsare determined by the conformal structure of the torus, so τ1 = τ2. This shows thatthe ratio |l2|/|l1| is monotone, and it is easy to check that it is increasing in Im τ . �
Lemma 4.5. There exists a continuous curve c : R → C such that τG ∈ c(R) andc(R)⊂ b−1(0).
Proof. Since b is continuous, b < 0 on Y and b > 0 on B, the zero set of b musttopologically separate B and Y ; in particular, it contains a curve c such that τG ∈ c.This curve does not intersect L or R, because there are no zeros on either. ThusIm(c(t))→ 0 or Im(c(t))→ ∞ as t → ±∞. �
Finally, the following proposition proves embeddedness.
Proposition 4.6. Let Mt for 0 ≤ t ≤ 1 be a continuously differentiable family of im-mersed triply periodic minimal surfaces. If M0 is embedded, then Mt is embeddedfor all 0 ≤ t ≤ 1.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 161
Proof. Let S be a fixed genus three Riemann surface. Then each Mt can beparametrized (not conformally) by a map ft : S → R3. We write the family ofsurfaces as a map f : S ×[0, 1] → R3 by f (x, t)= ft(x) and assume that this mapis continuously differentiable.
Let t0 be the first time that a surface is not embedded, that is,
t0 := inf{t > 0 | Mt is not embedded}.
We assume that t0 exists and arrive at a contradiction.We first prove that Mt0 is not embedded. Let tk → t0. Since f (S, tk) is not
embedded, there exists pk, qk ∈ S with pk 6= qk such that f (pk, tk) = f (qk, tk).Since S is compact, there is a convergent subsequence of pk and qk ; without lossof generality we relabel to obtain sequences pk → p and qk → q .
Case 1: p = q. Fix ε > 0. Since f ∈ C1 there exists δ1 > 0 and N > 0 such thatfor all k > N and for any x ∈ Bδ1(p),
(4-1) |N ( f (x, tk))− N ( f (p, t0)|< ε.
Since f (·, t) is an immersion for each t ∈ [0, 1], we know f (Br (p), tk) is anembedded minimal disk for sufficiently small r . Let
ηk = sup{r > 0 | f (Br (p), tk) is an embedded minimal disk}.
For sufficiently large k, we have pk, qk ∈ Bδ(p), and so f (Bδ(p), tk) is not em-bedded for k > N (possibly after increasing N ). Thus ηk < δ. Define rk =
ηk + (δ − ηk/2). Because f (Brk , tk) is not a graph over its tangent plane, thereexist two points whose orthogonal projections to the tangent plane are the same.Therefore, by the mean value theorem, there must be some point zk ∈ Brk (p) suchthat N ( f (zk, tk)) is parallel to the tangent space at p, which contradicts (4-1),provided ε is sufficiently small.
Case 2: p 6= q . Since f (p, t0) = f (q, t0) and f (S, t0) is embedded, for someδ > 0, we have f (Bδ(p), t0) = f (Bδ(q), t0). This implies that f (·, t0) : S → Mt0is a mapping of degree 2; this is a contradiction since both S and Mt0 have genusthree.
Having shown so far that Mt0 is not embedded, we know it has a point of self-intersection. Also, t0 > 0. Suppose that the planes tangent to the surface aretransverse at the intersection point. Since transversality is an open condition: bythe continuity of the family there exists an ε > 0 such that the tangent planesfor Mt0−ε will also be transverse. Therefore, Mt0−ε has a self intersection, whichcontradicts the minimality of t0. Thus the tangent planes must be coincident.
Suppose that f (p1, t0) = f (p2, t0) is a point of self intersection. As above,Br (pi ) is an immersed minimal disk Mi for sufficiently small r > 0. Each is aminimal graph over their common tangent plane. Define a height function hi on
162 ADAM G. WEYHAUPT
Mi as the height of graph(Mi ). By the maximum principle for minimal surfaces,we cannot have h1 −h2 > 0 on Br (p)−{p}. Thus h1 −h2 assumes some negativevalues. This being an open condition, there is an ε > 0 such that h1 − h2 is alsonegative on a neighborhood of G t0−ε . If h1 − h2 is both negative and positive,the surface G t0−ε can not be embedded since the two graphs M1 and M2 intersect,which contradicts the minimality of t0. Therefore, the family must be embeddedfor all t > 0. �
We can now prove the existence and embeddedness of the tG family.
Proof of Theorem 1.2. By Lemma 4.5, there exists a family of tori such that thedeveloped and normalized flat structures have the same vertical and horizontalturning angle θ . We use the Gauss map used to develop these flat structures, andset dh = eiθdz. This choice of height differential ensures that the flat structuresare in the moduli spaces V(G) and H(G). Therefore, the period problem is solvedfor this Weierstrass data. The branched torus cover provides the conformal modelof the triply periodic minimal surface, and the Gauss map G and dh that we havedefined lift, via the rotation ρ2, to a well-defined Gauss map and height differentialfor the triply periodic surface. This one-parameter family does contain the gyroid,by the description of the gyroid in Section 3.3. The entire family is embedded byProposition 4.6. �
5. The rG and rL families
5.1. Description of the Lidinoid. The H surface (Figure 3.1) is a genus threetriply periodic minimal surface that admits an order 3 rotational symmetry. It canbe thought of as containing a “triangular catenoid” in the same way that the Psurface contains “square catenoids”. Its lattice is spanned by a planar hexagonallattice (along with a vertical component), in contrast to the square planar lattice forthe P surface. We proceed analogously to the P surface and omit most details.
The order 3 rotation ρ3 : R3→ R3 descends on H/3 to a well-defined isometry
with 2 fixed points. By Abel’s theorem, there are only two possible locations forthe pole of the Gauss map on the torus, 1/2 or 1/2+τ/2. Since there is a reflectivesymmetry of H whose fixed point set contains both fixed points of the map, thesame must be true on the torus. This forces the pole to be located at 1/2. Again,symmetry considerations force the torus to be rectangular, and we normalize sothat it is generated by 〈1, τ 〉 for τ ∈ iR. The zero and poles are second-order by acomputation similar to Lemma 2.9.
We write G3(z) := ρθ211(z, τ )/θ
211(z − 1/2, τ ) and we again set the Lopez–Ros
factor ρ = 1 for the appropriate normalizations.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 163
A1B1 A3
to 2 to 3
Sheet 1 A3A2
B2
to 3 to 1
Sheet 2
A3
B3
to 1 to 2
Sheet 3
Figure 5.1. The conformal structure of the H surface branchedcover of a torus. Cuts are shown by dashed lines.
The torus is invariant under −id , which is here the hyperelliptic involution.Figure 5.1 gives the conformal structure of the branched cover and the relevantcycles.
5.1.1. Flat structures. The 1-forms G dh, (1/G)dh, and dh each place a flat struc-ture on the torus which, after taking the quotient with −id, descends to the sphere.We study each flat structure for the H surface independently:
The dh flat structure. Since this descends as dz, the flat structure for the torus issimply the rectangle.
The G dh flat structure. Again, the order of the zeros and poles of the 1-form G dhproduce cone angles on the torus of 10π/3i at 0 and of 2π/3 at 1/2. The remainingfixed points, at τ/2 and 1/2 + τ/2, are regular points. The involution −id halvesthe cone angles in the quotient, so that on the sphere have cone points of angle10π/6 at 0, of angle π/3 at 1/2, and of angle π at τ/2 and 1/2 + τ/2.
The flat structure is a hexagon:
Lemma 5.1. By cutting along the shortest geodesics on the sphere from τ/2 to 0,0 to 1/2, and from 1/2 to 1/2+τ/2, we obtain a hexagon shown in black in Figure5.2. The hexagon has the properties that
(i) the length of li is equal to that of l∗i for i = 1, 2, 3;
(ii) the angle between l1 and l2 and the angle between l∗1 and l∗2 are both 5π/6;
(iii) the angle between l1 and l∗1 is π/2 and the angle between l3 and l∗3 is π .
The proof is precisely analogous to Lemma 3.1. We find this flat structureinconvenient, given that the flat structure on the entire torus (without the −ididentification) is so simple. The flat structure on the entire torus is obtained by
164 ADAM G. WEYHAUPT
(0, 0)
(12p,
√3
2 p)
(12p,
√3
2 p + q)
(0,√
3p + q)
(−12p,
√3
2 p + q)
(−12p,
√3
2 p)
Figure 5.2. The G dh (thick line) and (1/G)dh (thin line) flatstructures for the H surface. Labeled vertices are for the G dh flatstructure (the corresponding points on (1/G)dh are obtained bytranslation by (−1/2p,−
√3/2p).
rotating by π about the vertex between l3 and l∗3 (the −id map descends on thedeveloped image to the −id map since 1/2+τ/2 is a regular point). Doing so, weobtain the flat structure shown in Figure 5.2; compare to the P surface flat structure.
Again we have not chosen the imaginary part of τ as we expect to recover afamily of surfaces.
We now describe the (1/G)dh flat structure for the H surface. By preciselythe same argument as in Section 3.1.3, it is simply a translate of the G dh flatstructure, with the Lopez–Ros factor ρ = 1. The blue outline in Figure 5.2 showsthis translation.
5.1.2. The period problem for the H surface. We omit the computation of theperiods, but we obtain
P(A1)= (0, 0, 0),
P(A2)= (0, 0, 0),
P(A3)= (0, 0, 0),
P(B1)= (0,−√
3p − 2q, 0),
P(B2)= R2π i/3(0,−√
3p − 2q, 0),
P(B3)= (0, 0, 1).
Here the notation R2π i/3 means a rotation by 2π i/3 about the x3-axis.It is again immediate that these periods generate a 3-dimensional lattice3 for all
values of p and q. In other words, the period problem is solved no matter what theactual lengths of the segments in the developed flat structure are. Thus any valueof a (and therefore, any quotient torus) solves the period problem. This proves thatthe H surface comes in a 1-parameter family; in fact, the H surface is in Meeks’
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 165
family, so it comes in a 5-parameter family. This family is called the rH family.In the limit τ → 0, the rH surface looks like a pair of parallel planes joined withsmall catenoidal necks. As τ → ∞, it looks like three intersecting planes that aredesingularized along the intersection by adding handles. (Compare this to the tPfamily.)
To construct the Lidinoid, let (X,G, dh) be the Weierstrass data describing amember of the rH family. As in the case of the gyroid, we can easily calculatethe periods for all members of the associate family by using the H surface flatstructures. For instance, one can compute that for all 0 ≤ θ ≤ 2π , the associatesurface has periods
P(A1)= (√
3p sin θ, p sin θ,−−) and P(B3)= (−√
3p sin θ, p sin θ,−−).
Since these two periods clearly generate the horizontal part of the lattice, we mustensure the others are compatible. For instance,
P(B1)= (√
3p sin θ,−(2q +√
3p) cos θ,−−).
Thus, since sin(θ) 6= 0 for nontrivial members of the associate family,
±(2q +√
3p) cos θ = sin θ.
Examining the periods for B2 shows that we must choose the “+” equation, so that
(5-1) θ = arctan(−
√3p−2q
p
).
Similarly to the gyroid, Equation (5-1) puts a constraint on θ , and the verticalperiod condition places another condition; these two conditions are compatible forexactly one value of θ — the value that gives the Lidinoid.
The full set of periods of the Lidinoid for our parametrization are
P(A1)= R2π i/3(0,−1, 0),
P(A2)= −R2π i/3(0,−1, 0),
P(A3)= (0, 0, 3s),
P(B1)= −R4π i/3(0,−1, s),
P(B2)= −R4π i/3(0,−1, s),
P(B3)= R4π i/3(0,−1, s),
where s ∈ R+ is calculated with an elliptic integral.
5.2. Moduli spaces for the rL family.
5.2.1. Vertical moduli space V(L). The vertical moduli space is defined in pre-cisely the same way as for the gyroid, that is, V(L)= V(G).
166 ADAM G. WEYHAUPT
Figure 5.3. The Lidinoid.
5.2.2. Horizontal moduli space H(L). Suppose that M is any immersed, genusthree, triply periodic minimal surface that has as a conformal model a three-foldbranched cover of a generic torus C/0. Without loss of generality we write0 = 〈1, τ 〉. Suppose further that the square of the Gauss map descends to C/0
and has a second-order pole at 1/2 and a second-order zero at 0. (This is the casefor the Lidinoid, except that the torus is rectangular.) The quotient S = C/0/− idis a sphere, and G dh again induces a cone metric on S. Under this cone metric,the sphere is a tetrahedron, with vertex angle of 5π/3 corresponding to the zero, avertex angle of π/3 corresponding to the pole, and two vertex angles of π corre-sponding to the remaining fixed points τ/2 and 1/2 + τ/2 of −id . The developedimage of this sphere is parametrized as follows.
Lemma 5.2. For any torus C/0 with 0 = 〈1, τ 〉, the cone metric G dh descendsto S. By cutting along shortest geodesics on S from 1/2 to 0, 0 to τ/2, and from τ
2to 1
2 +τ2 , we obtain a hexagon. The hexagon has the properties that
(i) the length of li is equal to that of l∗i for i = 1, 2, 3;
(ii) l2 is parallel to l∗2 ;
(iii) the angle between l1 and l∗1 and the angle between l3 and l∗3 are both π/2.
Since the fixed point 1/2 + τ/2 is regular before the application of −id , wecan extend this to a developing map on the whole torus by rotating by π about theintersection of l4 and l∗4 . Doing this, we obtain the hexagon flat structure shown inFigure 5.4. We can parametrize this final space of possible hexagons by ξ1, ξ2 ∈ C
as shown in Figure 5.4.
Let H(G) be the space of all hexagons satisfying the conditions of Lemma 5.2.The proof is precisely analogous to that of Lemma 4.1, and we omit the details.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 167
l1
l∗1
l3 l∗3
π
3
π
3
0
e−iπ
3 ξ1
ξ1
ξ2
ξ2 − e−iπ
3 ξ1
ξ2 + e−iπ
3 ξ1
Figure 5.4. A generic member of the H(L) moduli space.
To understand the horizontal periods, we again adjust ρ, if necessary, to nor-malize the (1/G)dh flat structure as in Section 3.1.3.
After the substitutions
c = −12(Im ξ1 +
√3 Re ξ1) and d =
12(Im ξ1 − 4 Im ξ2 +
√3 Re ξ1),
the periods can be expressed as
P(A1)= (√
3c, ci, 0),
P(A2)= (−√
3c,−ci, 0),
P(A3)= (0, 0, 3s),
P(B1)= (√
3c, d, s),
P(B2)= (−(√
3/2)(d − c),−(1/2)(d − c), s),
P(B3)= (−√
3c, c, s).
Again, s is a factor determined by the torus. Since P(B1)= P(B2) for the Lidinoid,we are forced to set c = −d to solve the period problem. Therefore, the periodproblem is solved if Im ξ2 =0. Note that, apart from parametrizing the flat structuredifferently (here 0 corresponds to a different cone angle on the P surface), this isprecisely the same condition as for the gyroid flat structures.
The same argument as before proves Theorem 1.3.
5.3. Description of the gyroid from the standpoint of the rPD family. In additionto the order 2 rotational symmetry, the standard, most symmetric P surface alsoadmits an order 3 symmetry. This symmetry permutes the handles of the P surfaceand is obtained by rotating by 2π/3 though the normal at one of the eight pointswhere the Gaussian curvature K = 0. We repeat the procedure discussed above forthe P surface (viewed as invariant under an order 2 rotation), and we obtain again
168 ADAM G. WEYHAUPT
a one-parameter family of P surfaces, which this time is invariant under an order3 rotation.
Since the standard P surface is a member of the rPD family, the gyroid can alsobe parametrized in terms of it. We outline the construction of the gyroid in thisway, so that we can construct a second family of gyroids, the order 3 gyroids rG.
To begin, we need to find the conformal parameter τ that yields the standard,most symmetric P surface. From the end of Section 3.3, recall that the standard Psurface can be described in terms of the 1-forms ω1, ω2, and ω3; these forms areconsidered with the orientation of the P surface in space so that the lattice is thestandard, cubical lattice. These are permuted by the rotation ρ3. After a rotationof the surface in space so that the axis of rotation is vertical, dh = ω1 +ω2 +ω3.We understand the periods of these 1-forms explicitly from our work with theP surface. Denote by γ1 the cycle generated by the vector 1 on the order 2 Psurface torus. Its period on each of the ωi flat structures is 1, that is,
∫γ1ωi = 1 so
that∫γ1
dh =∫γ1ω1 +ω2 +ω3 = 1. This implies that one generator of the quotient
torus P/ρ3 is 1, since the location of the branch cuts implies that this cycle γ1
continues onto all three sheets.Considering the other generator of the torus (rather, the cycle γ2 coinciding with
this generator), we note that∫γ2ωi = 2a, since the cycle continues onto both sheets
of the torus P/ρ2 (recall that a = Im τ ≈ 0.78). On the other hand, if we denotethe generators of the torus P/ρ3 by 1 and σ , then
∫γ2
dh =∫γ2ω1 +ω2 +ω3 = 6a
but also∫γ2
dh = 3 Im σ since the cycles continues onto both sheets. Thus, thestandard P surface is obtained when σ = 2τ . Since this is the standard P surface,the angle of association that yields the gyroid is the same θ = arccot Im τ . This isprecisely the same surface as obtained in Section 3.3, but viewed from a differentperspective and using a different parametrization. Another view of the gyroid is inFigure 5.5.
5.4. Moduli spaces for the rG family.
5.4.1. Vertical moduli space V(rG). To obtain the standard gyroid from the order3 perspective, we take as the torus parameter τ = 2 · ai , where a is the conformalparameter for the standard P surface. We then use the same angle of associationas in the order 2 parametrization, obtaining, Re eiθ
= −2 Re eiθτ . We thereforedefine V(rG) = {(ω1, ω2) ∈ C × C | |ω1| = 1 and Reω1 = −2 Reω2}, so that thevertical period problem is solved.
5.4.2. Horizontal moduli space H(rG). Suppose M is an immersed, genus three,triply periodic minimal surface that has as a conformal model a three-fold branchedcover of a generic torus C/0. Without loss of generality we write 0 = 〈1, τ 〉.Suppose further that the square of the Gauss map descends to C/0 and has asecond-order pole at 1/2 + τ/2 and a second-order zero at 0. (This is the case
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 169
Figure 5.5. Left: a translational fundamental domain of the gy-roid, viewed as invariant under an order 3 rotation. Right: severalcopies of a fundamental domain. Notice the similarity to the Lidi-noid.
for the order 3 gyroid, except that the torus is rectangular.) The quotient S =
C/0/− id is a sphere, and G dh again induces a cone metric on S. Under thiscone metric, the sphere is a tetrahedron, with vertex angle of 5π/3 corresponding tothe zero, a vertex angle of π/3 corresponding to the pole, and two vertex angles ofπ corresponding to the remaining fixed points τ/2 and 1/2 of −id . The developedimage of this sphere is described as follows.
Lemma 5.3. For any torus C/0 with 0 = 〈1, τ 〉, the cone metric G dh descendsto S. By cutting along shortest geodesics on S from 1/2 to 0, from 0 to τ/2, andfrom τ/2 to 1/2+τ/2, we obtain a hexagon shown in Figure 5.6. The hexagon hasproperties that
(i) the length of li is equal to that of l∗i for i = 1, 2, 3;
(ii) l2 = ei2π/3l∗2 ;
(iii) the angle between l1 and l∗1 is π and the angle between l3 and l∗3 is π/3.
We can parametrize this final space of possible hexagons by ξ1, ξ2 ∈ C as shownin Figure 5.6. We use the notation ξ1 = e−iπ/3(ξ2 − ξ1)+ ξ1 − e−2π i/3ξ1.
We call the space of all hexagons satisfying the conditions of Lemma 5.3 H(rG).To understand the horizontal periods, we again adjust ρ, if necessary, to nor-
malize the (1/G)dh flat structure as in Section 3.1.3. Then in terms of these flat
170 ADAM G. WEYHAUPT
ξ1
ξ2
ξ1
0
e−iπ/3ξ1
α
α∗
π3
l3
l∗3
l2
l1
Figure 5.6. A generic member of the H(rG) moduli space.
structures, we compute the periods of the six generators of H1(M, Z) to be
P(A1)= (a, 0,−s),
P(A2)= (a, 0,−s),
P(A3)= (a, 0,−s),
P(B1)= (b,√
3(a − b), 2s),
P(B2)= (b,√
3(a − b), 2s),
P(B3)= (b,√
3(a − b), 2s).
Here we simplified the expressions using the substitutions
a = 2√
3 Im ξ1 −√
3 Im ξ2 + 2 Re ξ1 − 3 Re ξ2,
b =√
3 Im ξ1 − (√
3/2) Im ξ2 + Re ξ1 − (5/2)Re ξ2.
Again, s is a factor determined by the torus. Since, for the gyroid, one computesthat P(A1) = P(B2), we must set a = b to solve the period problem. Therefore,the period problem is solved if
√3 Im ξ1 − (
√3/2) Im ξ2 + Re ξ1 − (1/2)Re ξ2 = 0.
This seemingly complicated expression is actually very reasonable; it holds if andonly if arg
(ξ1 − ξ2
)= π/3. But the consequence of this is that any member of
H(rG) solves the period problem after a rotation (and so we can again define therelative turning angles). We define
H(rG)={(ξ1, ξ2) ∈ H(rG)
∣∣√3 Im ξ1 − (√
3/2) Im ξ2 +Re ξ1 − (1/2)Re ξ2 = 0}.
The rest of the proof of the existence of a family of order 3 gyroids is analogousto the other two families discussed above, and this proves Theorem 1.4.
DEFORMATIONS OF THE GYROID AND LIDINOID MINIMAL SURFACES 171
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[Farkas and Kra 1992] H. M. Farkas and I. Kra, Riemann surfaces, Second ed., Graduate Texts inMathematics 71, Springer, New York, 1992. MR 93a:30047 Zbl 0764.30001
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[Große-Brauckmann and Wohlgemuth 1996] K. Große-Brauckmann and M. Wohlgemuth, “The gy-roid is embedded and has constant mean curvature companions”, Calc. Var. Partial DifferentialEquations 4:6 (1996), 499–523. MR 97k:53011 Zbl 0930.53009
[Karcher 1989] H. Karcher, “The triply periodic minimal surfaces of Alan Schoen and their con-stant mean curvature companions”, Manuscripta Math. 64:3 (1989), 291–357. MR 90g:53010Zbl 0687.53010
[Lidin and Larsson 1990] S. Lidin and S. Larsson, “Bonnet transformation of infinite periodic min-imal surfaces with hexagonal symmetry”, J. Chem. Soc. Faraday Trans. 86:5 (1990), 769–775.
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[Meeks 1975] W. H. Meeks, III, The Geometry and the Conformal Structure of Triply PeriodicMinimal Surfaces in R3, PhD thesis, University of California, Berkeley, 1975.
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[Schoen 1970] A. H. Schoen, “Infinite periodic minimal surfaces without self-intersections”, Tech-nical note TN D-5541, NASA, 1970.
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Received May 31, 2007. Revised July 6, 2007.
ADAM G. WEYHAUPT
DEPARTMENT OF MATHEMATICS AND STATISTICS, BOX 1653SOUTHERN ILLINOIS UNIVERSITY
EDWARDSVILLE, IL 62026UNITED STATES
[email protected]://www.siue.edu/~aweyhau
PACIFIC JOURNAL OF MATHEMATICSVol. 235, No. 1, 2008
AN EXTENSION PROCEDUREFOR MANIFOLDS WITH BOUNDARY
JEREMY WONG
This paper introduces an isometric extension procedure for Riemannianmanifolds with boundary, which preserves some lower curvature boundand produces a totally geodesic boundary. As immediate applications ofthis construction, one obtains in particular upper volume bounds, an upperintrinsic diameter bound for the boundary, precompactness, and a home-omorphism finiteness theorem for certain classes of manifolds with bound-ary, as well as a characterization up to homotopy of Gromov–Hausdorfflimits of such a class.
1. Introduction
For manifolds with boundary, there seems to be a strong connection between theexistence of a curvature-controlled extension and uniqueness or finiteness results.
For example, in proving a uniqueness theorem for minimal surfaces, Nitsche[1973] used an extension procedure to extend a minimal surface with boundary afixed distance beyond its boundary (as a minimal surface). According to a theoremof Lewy [1951], if a minimal surface in R3 (possibly with interior branch points)has a boundary consisting of real analytic boundary curves, then the surface canbe extended beyond its boundary as a minimal surface.
Motivated by the Penrose conjecture, Bartnik [1993] considered the extensionproblem in the class of positive-mass metrics. This problem states, given a boundedRiemannian three-manifold, describe the class of complete three-manifolds satis-fying the conditions of the positive mass theorem (in particular, asymptotic flatnesswith nonnegative scalar curvature) and containing the original three-manifold iso-metrically. By solving a parabolic PDE, he established a special case of the Pen-rose conjecture, assuming in particular a foliation of 3-space by metric 2-spheresof positive Gaussian curvature, and boundary data being a minimal surface. Theconclusion was that the Schwarzschild metric was distinguished as having the leasttotal mass among all such 3-metrics.
MSC2000: primary 53C20, 53C21; secondary 51K10.Keywords: manifold with boundary, extension, Gromov–Hausdorff topology.The author was supported in part by NSF fellowship.
173
174 JEREMY WONG
Thus, it is important to find general isometric, curvature-controlled extensionprocedures.
The implicit function theorem and the Cauchy–Kowalevski theorem (in the an-alytic category) are common tools to extend a manifold with boundary. However,not only do these have a short, if not merely infinitesimal extension range, theseprocedures are also, unlike the methods of the results mentioned above, insensitiveto curvature constraints.
In the context of Riemannian manifolds with nonnegative sectional curvatureand locally convex boundary, Kronwith [1979] considered C2 extensions preserv-ing nonnegative sectional curvature and locally convex boundary, but his approachrelied on power series expansions of the metric tensor and features special to two-dimensional surfaces which do not seem to work for dimensions higher than two.
Whitney extension of the metric tensor coefficients is another viable procedure,but to control sectional curvature of such an extension, one must assume a bound onthe first two covariant derivatives of the curvature tensor of the original manifold.This is generally regarded as too strong a geometric assumption to make. If oneis not interested in isometric extensions, meaning extensions leaving the metrictensor on the original manifold intact, then it is possible to combine the Whitneyextension with various smoothing techniques to guarantee that the higher-ordercovariant derivatives of the curvature tensor are bounded.
Normal extension techniques, such as used in [Kim et al. 2005], do provide anypreselected uniform extension range, though so far their use seems largely confinedto two-dimensional surfaces. This technique leads to focal points, which explainswhy a lower bound on the second fundamental form of the boundary must typicallybe assumed.
Here, we introduce an isometric extension procedure for manifolds with bound-ary of any dimension; this procedure preserves some lower curvature bound andproduces a totally geodesic boundary.
Beginning with any Riemannian manifold with boundary (M, ∂M) one maymanufacture a collar, which, when isometrically glued to the boundary, yields anAlexandrov space of curvature bounded below. Outside the gluing locus ∂M , theresulting extension M is C∞. Actually, M is a C0 Riemannian manifold with aC1,α differentiable manifold structure. An important feature is that not only is thecurvature of M bounded from below, but M can be constructed so as to have atotally geodesic boundary.
This (Alexandrov) extension procedure — so-called since the result is an Alex-androv space — has several applications. Among these, for certain classes of man-ifolds with boundary, are an upper volume bound, an upper bound to the numberof boundary components, an upper intrinsic diameter bound to any boundary com-ponent, a dimension estimate for Gromov–Hausdorff limits, precompactness, a
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 175
homeomorphism finiteness result, and a characterization up to homotopy of limits.Further applications are given in [Alexander et al. ≥ 2008], for topological
finiteness of all possible locally convex hypersurfaces in Rn spanning a givencodimension-2 smooth submanifold, and in [Wong 2007], for studying collapsesof manifolds with boundary.
Now we state the main results of this paper more precisely.Fix n ≥ 2, K −, λ±, and d ∈ R. Let M(n, K −, λ±, d) denote the class of n-
dimensional Riemannian manifolds with boundary with lower sectional curvaturebound KM ≥ K −, two-sided second fundamental form bound λ−
≤ II∂M ≤ λ+, andupper intrinsic diameter bound d(M)≤ d . Let M ∈ M(n, K −, λ±, d).
Theorem 1.1. (i) vol(M)≤ V for some V = V (n, K −, λ±, d), a universal posi-tive constant.
(ii) vol(∂M)≤ V for some constant V = V (n, K −, λ±, d) <∞.
(iii) The intrinsic diameter of any boundary component of M is uniformly boundedabove by d(∂M)≤ D(n, K −, λ±, d).
(iv) ∂M has no more than c components, where c = c(n, K −, λ±, d) is a finiteconstant.
Next, the class M may be endowed with the Gromov–Hausdorff topology.There is a corollary to the upper volume estimate given above.
Corollary 1.2. If a sequence Mi ∈ M(n, K −, λ±, d) GH-converges to a metricspace X (necessarily compact and geodesic) then the Hausdorff dimension of Xsatisfies dimH X ≤ n.
Relative volume comparison does not hold for the class M(n, K −, λ±, d) norfor a more restricted class in which one imposes a lower volume bound to themanifold and an intrinsic injectivity radius bound for the boundary. The ratio ofthe volume of a ball to the volume of a smaller subball may have no upper bound.Consider for instance a neckpinch, in which two concave parts of the boundary arearbitrarily close to each other. Precompactness does nevertheless hold for theseclasses, which is remarkable since relative volume comparison (within the classitself) is customarily used to prove it.
Theorem 1.3. M(n, K −, λ±, d) is precompact in the Gromov–Hausdorff topology.
In other words, any sequence of manifolds with boundary in this class contains asubsequence that converges to a compact metric space.
Theorems 1.1(iv) and 1.3 suggest that the class considered there contains a sub-class for which there are only finitely many topological types. Indeed, their numberis bounded if a lower volume bound is also imposed:
176 JEREMY WONG
Theorem 1.4. M(n, K −, λ±, vol ≥ v > 0, d) contains only finitely many homeo-morphism classes.
This is the analogue in the bordered case of the Grove–Petersen–Wu theorem.The extension procedure also allows one to deduce homotopy structure for limits
of certain manifolds with boundary.
Theorem 1.5. Suppose {Mi } is a sequence of n-dimensional Riemannian mani-folds with boundary such that KMi ≥ K −, |II∂Mi | ≤ λ, d(Mi ) ≤ d , and Mi GH-converge to a limit space. Then there is a Lipschitz homotopy equivalence
limGH Mi ' limGH Mi ,
where Mi are the (Alexandrov) extensions of Mi , as in Proposition 2.1.
In particular, because limGH Mi is homotopic to the locally contractible spacelimGH Mi , it is itself locally contractible and hence admits a universal cover.
The next result is the main aid to constructing the extension.
Theorem 1.6 [Kosovskiı 2002]. Let M1 and M2 be two smooth Riemannian man-ifolds with boundary. Suppose their sectional curvatures are bounded below by Kand their boundaries are isometric and have respective second-fundamental forms,the sum of which is positive semidefinite. Then the space obtained by isometricallygluing M1 to M2 along their common boundary is an Alexandrov space of curva-ture bounded below by K .
Another essential element in the construction of the extension is the devisingof a collar with the right curvature properties, which will assume the role of M2
above.
Lemma 1.7. Suppose M is any manifold with boundary having KM ≥ K − andλ−
≤ II∂M ≤λ+. Then for any t0>0, there exists an intrinsic metric on ∂M×[0, t0]such that II∂M×{0} ≥ |min{0, λ−
}| and II∂M×{t0} ≡ 0 and the sectional curvature of∂M × [0, t0] is bounded below by a constant c(K −, λ±, t0).
Letting CM = ∂M × [0, t0] denote the collar so produced, the extension M isdefined as
M = M⋃∂MCM .
The metric on CM is constructed so that ∂M ×{0} in CM has a degree of convexitywhich is at least as great as any possible concavity of ∂M in M . The properties ofthe extension M are given in more detail in Section 2.1.
By combining Lemma 1.7 with a similar, but different, gluing technique forRicci curvature, Theorems 1.1 and 1.3 may be improved.
Theorem 1.8 [Perelman 1997]. Let M1 and M2 be two C∞ Riemannian manifoldswith compact isometric boundaries, each having ric(Mi )>r− for some r−
∈R, and
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 177
boundaries satisfying II∂M1 + II∂M2 ≥ 0 at every point on the identified boundary.Then for all δ > 0 sufficiently small (depending on M1 and M2) the induced metricon M1 ∪ M2 may be smoothed in a δ-small neighborhood of the gluing locus ∂Mi
to a C2 manifold with ric(M1 ∪ M2) > r−.
Originally Theorem 1.8 was sketched in [Perelman 1997] for the case of posi-tive Ricci curvature and positive definite sum of second fundamental forms. Thedetails were provided in [Wang 1997]. It is not hard to check that this generalizesto any lower Ricci curvature bound, as long as the sum condition on the secondfundamental forms is satisfied.
Let M(n, r−, λ±, d) denote the class of n-dimensional Riemannian manifoldswith boundary with lower Ricci curvature bound ric(M) ≥ r−, two-sided secondfundamental form bound λ−
≤ II∂M ≤ λ+, and upper intrinsic diameter boundd(M)≤ d. Let M ∈ M(n, r−, λ±, d).
Theorem 1.9. (i) vol(M)≤ V (n, r−, λ±, d).
(ii) ∂M has no more than c components, where c = c(n, r−, λ±, d) is a finiteconstant.
(iii) The intrinsic diameter of any boundary component of M is uniformly boundedabove by d(∂M)≤ D(n, r−, λ±, d).
(iv) M(n, r−, λ±) is precompact in the pointed Gromov–Hausdorff topology.
This theorem is fairly sharp, in that counterexamples arise if any of the hypothe-ses, except perhaps the upper bound λ+, are dropped. The lower bound λ− (or atleast a lower mean curvature bound H− put in its place) is necessary for (i), (iii),and (iv) to hold, as examples show. See Figure 1.
For another example, consider removing from the standard unit sphere arbitrarilymany small disjoint balls. Thus, the bound λ− is also necessary for part (ii) ofTheorem 1.9.
arbitrarily negative II∂M
Figure 1. A sunflower with arbitrarily many petals.
178 JEREMY WONG
When n = 2, Lemma 1.7 — and hence Theorems 1.1–1.5 and Theorem 1.9 —remains true if the upper bound λ+ is dropped.
This paper is organized as follows. After a few additional remarks putting themethods and results into context, Section 2.1 constructs the extension. The mainresult is Proposition 2.1. Section 2.2 proves Theorems 1.1–1.5 and Theorem 1.9.
Remarks on gluing
The key assumption of positive semidefiniteness of the sum of the second funda-mental forms of the boundaries in the gluing procedure of Kosovskiı [2002] andPerelman [1997] has also been used by Miao [2002]. He obtained a metric ofbounded scalar curvature on the union of two manifolds with boundary, assumingthat each manifold had bounded scalar curvature and that the boundaries satisfiedthe sum condition for mean curvature. See [Miao 2002, Proposition 3.1] for detailsand the proof. The sum condition for mean curvature was proposed in print in[Bartnik 1997]. Recently it has been used to investigate versions of the positivemass theorem [Miao 2002; Shi and Tam 2004].
However, the gluing theorem for Ricci curvature, Theorem 1.8, will not hold ingeneral if the sum condition on the second fundamental forms is replaced by a sumcondition on the mean curvatures. See [Wei 1989, page 19] for a counterexample.
One of the earliest references to a sum condition is [Alexandrov 1948], translatedin [Aleksandrov and Zalgaller 1967], in which the gluing of several domains withboundary — each cut out of a two-dimensional space of nonnegative curvature (bywhich is meant either a convex polyhedron, a convex surface, or a C∞ Riemannianmanifold) and having sum of turns ≥ 0 — itself was a space of nonnegative cur-vature. The notion of turn is a generalization of the integral of geodesic curvatureand coincides with it when the boundary is at least C2 smooth.
If one is interested in studying a class of manifolds with boundary via thetechnique of gluing, it is necessary to produce a compatible gluand to attach toa boundary. If this is done smoothly, this is the Cauchy extension problem.
Remarks on [Kodani 1990]
Theorems 1.3 and 1.4 amend and sharpen certain results of Kodani, whose [1990]paper was one of the earliest to consider convergence of Riemannian manifoldswith boundary with respect to the Gromov–Hausdorff topology.
Theorem 1.10 [Kodani 1990]. In M(n, K ±, λ±, i∂ , iint), the Gromov–Hausdorffand Lipschitz topologies coincide, that is, for every δ > 0 there exists an ε > 0 suchthat for all M, N ∈ M, dG H (M, N ) < ε implies dL(M, N ) < δ.
Here, the assumed uniform lower bounds on the injectivity radii i∂ and iint
provide a uniform lower volume bound to any M ∈ M. Refer to Appendix A
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 179
for the definitions of these injectivity radii. Theorem 1.10 is a generalization ofGromov’s convergence theorem (see [Gromov 1999, Chapter 8+, Section D] andits references), which asserts that for any sequence of closed n-dimensional Rie-mannian manifolds Mi satisfying |KMi | ≤ K , d(Mi ) ≤ d , and vol(Mi ) ≥ v > 0,there is a subsequence that convergences in Lipschitz distance to an n-dimensionaldifferentiable manifold with a metric of C1,α Holder class.
Kodani gave precompactness results for two classes of manifolds with boundary.The first class involved manifolds with locally convex boundary:
Proposition 1.11. M(n, K ±, λ−= 0, λ+, d, vol ≥ v) is contained in the class of
Theorem 1.10. Furthermore, M(n, K ±, λ−= 0, λ+, d, vol ≥ v) is precompact in
the Gromov–Hausdorff topology.
Kodani also states, but with a circular proof1, the following proposition:
Proposition 1.12. There exist λ−
0 < 0 and i0 > 0 depending on n, K ±, λ+, d,vol ≥ v, and d∂ , such that i∂(M)≥ i0 and iint(M)≥ i0 is satisfied by any manifoldwith boundary M in the class M(n, K ±, λ−
0 , λ+, d, vol ≥ v, d∂), and so M is con-
tained in the class of Theorem 1.10. The class M(n, K ±, λ−
0 , λ+, d, vol ≥ v, d∂) is
precompact in the Gromov–Hausdorff topology.
Here the bound d∂ denotes a fixed upper bound for the sum of the intrinsicdiameters of all components of the boundary. In Proposition 1.12, some concavityof the boundary is allowed, but not too much (depending on the other geometricbounds).
By a result of Shikata [1966], if the Lipschitz distance dL(M, N ) is sufficientlysmall for two differentiable spaces M and N , then they are diffeomorphic. SoTheorem 1.10 implies
Corollary 1.13 [Kodani 1990]. The classes in Propositions 1.11 and 1.12 containonly finitely many diffeomorphism classes. In particular, the number of connectedcomponents of the boundary of any manifold in such a class is uniformly boundedabove.
The method used to prove Theorem 1.10 consisted of studying inward equidis-tant parallels of the boundary, relying on the i∂ bound.
1[Kodani 1990, Proposition 6.2], used to prove Proposition 1.12 above, relies on [Lemma 3.4],which requires the geodesic emanating from a point on the boundary to lie within an i∂ tubularneighborhood of the boundary. It may happen that in [Proposition 6.2] the minimizing M-geodesicτ2 (which begins on ∂M) goes outside such a neighborhood, so that τ2 * M[0,min{t0, iint, i∂ }]. Inthis case, the differential inequality used to derive an upper bound for l(s) := d(τ2(s), ∂M) is onlyvalid for certain 0 ≤ s ≤ s1, such that τ2(s) lies within the i∂ neighborhood of ∂M . So the upperbound on l(s) holds only if an upper bound on l(s) holds, which is a circular argument invalidating[Proposition 6.2] and thereby the proof of Proposition 1.12 given in [Kodani 1990].
180 JEREMY WONG
Examples show that Theorem 1.10 is sharp, in that none of the bounds may bedropped. On the contrary, Propositions 1.11 and 1.12 and Corollary 1.13 above arefar from optimal, as Theorems 1.3 and 1.4 show. In particular, for precompactness,it is not necessary to assume either an upper bound on the sectional curvature or alower bound for volume.
Notations and conventions
Manifolds are assumed to be metrically complete and also, unless we are speakingabout the boundary, usually connected. A closed manifold is one that is compactand without boundary.
For an immersion N ↪→ M , an inequality of the form II ≥ λ signifies that alleigenvalues of the associated quadratic form S : T N → T N are ≥ λ. Here wedefine II(X, Y ) = g(∇Xν, Y ), where ν is the outer normal. By convention, thestandard flat disc D2(r) of radius r has II = 1/r ≥ 0 and a convex boundary.
If N is a disconnected Riemannian manifold, an inequality of the form d(N )≤dwill usually be interpreted to mean that every path component of N has an upperintrinsic diameter bound d . Finally, we adopt these notations:
• [xy]X , for a length space X , is a minimal geodesic segment from the point xto the point y.
• B(x, r; X) is an open metric ball in X of radius r centered at x .
• dX , d, or |·|X interchangeably denote the metric distance function of a metricspace X .
• M(n, K ±, λ±, iint, i∂ , d) for instance, denotes the class of n-dimensional man-ifolds with boundary M having lower (K −) and upper (K +) interior sectionalcurvature bounds, lower (λ−) and upper (λ+) bounds on the second funda-mental form, some (unspecified) uniform positive lower bounds to iint(M) andi∂(M), and an upper diameter bound d .
2. (Alexandrov) extension
2.1. Construction.Proof of Lemma 1.7. Let λ = min{0, λ−
}. Fix some t0 > 0 and 0 < ε < 1. Forsome K = K (λ, ε, t0) ∈ R, there exists a C∞ monotone nonincreasing functionφ(t) defined on [0, t0] satisfying (see Figure 2)
φ′′+ Kφ ≤ 0, φ(0)= 1,
−∞< φ′(0)≤ λ,
φ(t0)= ε,
φ′(t0)= 0.
Consider the warped-product metric on ∂M × [0, t0] given by g1(x, t)= dt2+
φ(t)2g∂M(x). Let X and Y be g1-orthonormal tangent vectors to ∂M × [0, t0] at
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 181
0
slope = λ−
1
ε
t0
Figure 2. Warping function φ.
the point (x, t) that are vertical with respect to the Riemannian submersion thatprojects (∂M ×[0, t0], g1) onto [0, t0]. Let T denote a radial (that is, a horizontal)unit tangent vector to ∂M × [0, t0] at (x, t). From the O’Neill formula for a Rie-mannian submersion, g1 by construction has tangential sectional curvatures andradial curvatures
K1(X, Y )=K∂M(X, Y )−|φ′(t)|2
φ2(t)and K1(X, T )= −
φ′′(t)φ(t)
≥ K ∈ R,
respectively. If X i and X j are any unit vertical tangent vectors,
IIX i ∧X j (x, 0)= −φ′(0)φ(0)
≥ |λ| and IIX i ∧X j (x, t0)= −φ′(t0)φ(t0)
= 0,
so that the boundary is locally convex as stated. �
In Lemma 1.7, t0 and ε are independent free parameters which may be chosenaccording to one’s purpose. The optimal (that is, the greatest) lower bound K0
achievable for some φ satisfying the above requirements decreases to −∞ as t0decreases to 0 (with ε fixed). It also decreases to −∞ as ε tends to 1 (with t0fixed), provided λ− < 0 is fixed too.
An explicit warping function φ. Here is an explicit construction of a warpingfunction φ satisfying the condition of the lemma.
Assume that 0<ε < 1, 0< t0, and λ≤ 0. It may be further assumed that λ< 0,since otherwise λ−
≥ 0, and then the boundary, being locally convex, would notrequire an extension (in this case one could just take φ ≡ 1).
For 0 ≤ t < t0, define
(2-1) φ(t)= (1 − ε) exp(λt2
0
1 − ε
( 1t0−t
−1t0
))+ ε.
182 JEREMY WONG
Then
φ′(t)=λt2
0
(t0 − t)2exp
(λt2
0
1 − ε
( 1t0−t
−1t0
)).
Extend φ to be defined on [0, t0] by requiring continuity of φ and all its deriva-tives, that is, φ(t0) := limt↑t0 φ(t), φ
′(t0) := limt↑t0 φ′(t), and so on.
Then
φ(0)= 1,
φ′(0)= λ,
φ(t0)= limt↑t0 φ(t)= (1 − ε) · 0 + ε = ε,
φ′(t0)= limt↑t0 φ′(t)= 0,
and obviously φ′(t)≤ 0 for all 0 ≤ t ≤ t0 (so φ is nonincreasing) since λ≤ 0. Now
φ′′(t)=
(2
λt20
(t0 − t)3+
(λ)2t40
(t0 − t)4(1 − ε)
)exp
(λt2
0
1 − ε
( 1t0−t
−1t0
))has four possible critical points 0, t0, and the two points
t3± = t0 +16(3 ±
√3)λt2
0
1 − ε,
with corresponding values φ′′(0)= 2λ/t0 +λ2/(1−ε), φ′′(t0)= limt↑t0 φ′′(t)= 0,
and
(2-2) φ′′(t3±)= ∓ 432
√3(1 − ε)3
λ2t40 (3 ±
√3)4
exp(
−6
3±√
3−λt0
1−ε
).
But t3± ∈ [0, t0] if and only if
(2-3)|λ|t01 − ε
≤6
3 ±√
3.
Observe that 1≥φ(t)≥ε>0 for all 0≤ t ≤ t0. If φ′′(t)<0 then −φ′′(t)/φ(t)≥0.If φ′′(t) > 0 then
(2-4) −φ′′(t)φ(t)
≥ −1ε
maxφ′′(t)= −1ε
max{φ′′(0), φ′′(t0), φ′′(t3±)}
= −1ε
max{
0,(
2 λt0
+λ2
1−ε
), φ′′(t3−)
}provided that (2-3) holds for both cases. (Even if they do not, the bound here isstill valid, though not as sharp.) Therefore in any case, the radial curvatures arebounded below according to (2-4).
It remains to bound the tangential curvatures from below.The derivative φ′(t) itself has at most three possible critical points 0, t0, and
t4 = t0 +λt20/(2(1−ε)) in [0, t0], with corresponding values φ′(0)= λ, φ′(t0)= 0,
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 183
and
(2-5) φ′(t4)= 4(1−ε)2
λt20
exp(
− 2 −λt0
1−ε
).
But note that t4 ∈ [0, t0] if and only if |λ|t0/(1 − ε)≤ 2.Now note
K −
∂M/φ(t2)≥ 0 if K −
∂M ≥ 0,
K −
∂M/φ(t2)≥ K −
∂M/ε2 if K −
∂M ≤ 0.
Therefore in any case, the tangential sectional curvatures are bounded below by
(2-6) 1φ2(t)
(K −
∂M − |φ′(t)|2)≥ min{0, K −
∂M/ε2} − max|φ′(t)|2/minφ2(t)
≥1ε2
(min
{0, K −
− (max{|λ−|, |λ+
|})2}− max{|λ|2, (value of (2-5))2
}).
If in an orthonormal frame the sectional curvatures are bounded (from below)on all frame two-planes, then the sectional curvatures are bounded (from below) onarbitrary two-planes. So (2-4) and (2-6) together prove that the sectional curvaturesof (∂M × [0, t0], g1) are bounded below by a constant c(K −, λ±, t0), as stated inLemma 1.7.
Proposition 2.1 (Construction of extension). Fix n ≥ 2 and K −, λ±∈ R. For any
M ∈ M(n, K −, λ±)≡ {M Riemannian n-manifold : K −≤ KM , λ
−≤ II∂M ≤ λ+
},there exists an isometric, uniform extension M of M that is an Alexandrov space ofcurvature bounded below by a constant k = k(K −, λ±). The extension is uniformin that the distance in M between ∂M and ∂ M is no smaller than a constant whichmay be chosen arbitrarily.
Proof. Choose some t0 > 0 and 1 > ε > 0. Construct M as follows. Let CM =
∂M × [0, t0], and equip it with the metric g1 of Lemma 1.7. Let M := M ∪ CM
be the isometric gluing of CM to M along their isometric boundaries. Now letK −
C = K −
C (K−, λ±) denote the lower sectional curvature bound of the collar pro-
duced in the lemma. Then by Theorem 1.6, M is an Alexandrov space of curvaturebounded below by min{K −, K −
C }. The last claim follows by construction of themetric in Lemma 1.7, since geodesics emanating orthogonally to ∂ M minimize thedistance to ∂ M at least as long as they remain in CM . �
We remark that when dim M = 2 the upper bound λ+ is not needed in Lemma1.7 nor in Proposition 2.1 and its corollaries, since in this situation there are notangential two-planes on which to speak of curvature.
Note also that under only the hypotheses of Proposition 2.1, neither M nor CM
need be a locally convex subset of M . For instance, to see that CM need not belocally convex in M , take M to be the result of cutting lengthwise (through the
184 JEREMY WONG
apex) a rounded-off cone with small cone angle, so that the resulting boundary ofM near the apex is totally geodesic, but elsewhere has some concavity.
Properties of M. By [Kosovskiı 2002], a gluing as in Theorem 1.6, such as the Mconstructed in Proposition 2.1, is realized as the limit of manifolds with metric ten-sors of class W 2,∞
loc that have lower curvature bounds approaching that of M . Thesein turn are constructed using Sobolev averaging of the metrics in a neighborhoodof the gluing locus. As a consequence, M is a C1,α differentiable manifold withC0 Riemannian structure and is almost everywhere C∞.
For future reference, we record these additional properties of the extension M :
(i) i∂(M)≥ t0.
(ii) d(M)≤ d(M)+ 2t0.
(iii) |xy|M ≤1ε|xy|M for all x, y ∈ M . In particular, d(M)≤
1εd(M).
(iv) ∂M =1ε∂ M . In particular, d(∂M)=
1εd(∂ M).
Properties (i) and (iv) are clear from construction. Define a map π : M → M by
(2-7) π(x)=
{x if x ∈ M,orthogonal projection of x onto ∂CM = ∂M if x ∈ CM .
Let x, y ∈ M . Then
|xy|M ≤ |xπ(x)|M + |π(x)π(y)|M + |π(y)y|M by the triangle inequality
≤ t0 + |π(x)π(y)|M + t0 since M ⊂ M
≤ d(M)+ 2t0,
which proves (ii). Property (iii) will be proved in the course of Lemma 2.2.The Lipschitz continuity (as well as the Lipschitz constant) of the projection
map π from M to M will be important in later applications.
Lemma 2.2. Let (M, ∂M) be fixed. Consider the projection map π : M → Mdefined above. Then for all x, y ∈ M , we have |π(x)π(y)|M ≤
1ε|xy|M .
Proof. Let γ :=[xy]M be a minimal M-geodesic from x to y parametrized on [0, 1],and consider an arbitrary partition 0 ≤ t1 ≤ . . .≤ tN ≤ 1 of [0, 1]. Conceivably, γmay cross ∂M infinitely many times.
If [γ (ti )γ (ti+1)]M ⊂ M then
(2-8) |π(γ (ti ))π(γ (ti+1))|M = |γ (ti )γ (ti+1)|M .
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 185
If [γ (ti )γ (ti+1)]M ⊂ CM then
(2-9) |π(γ (ti ))π(γ (ti+1))|M ≤ |π [γ (ti )γ (ti+1)]M |M
= |π [γ (ti )γ (ti+1)]CM |M
= |π [γ (ti )γ (ti+1)]CM |∂CM
≤1ε|γ (ti )γ (ti+1)|CM
=1ε|γ (ti )γ (ti+1)|M .
Otherwise [γ (ti )γ (ti+1)]M belongs neither to M nor CM . In this case, whenγ (ti ) ∈ CM , let t∗
i ∈ ∂M be such that [γ (ti )γ (t∗
i )]M ⊂ CM ; otherwise set t∗
i := ti .Similarly, if γ (ti+1) ∈ CM , let t∗∗
i ∈ ∂M be such that [γ (t∗∗
i )γ (ti+1)]M ⊂ CM ;otherwise set t∗∗
i := ti+1.Note that π(γ (t∗
i ))= γ (t∗
i ) and π(γ (t∗∗
i ))= γ (t∗∗
i ), so that
(2-10) |π(γ (t∗
i ))π(γ (t∗∗
i ))|M = |γ (t∗
i )γ (t∗∗
i )|M ,
just as in (2-8). Then
|π(γ (ti ))π(γ (ti+1))|M
≤|π(γ (ti ))π(γ (t∗
i ))|M +|π(γ (t∗
i ))π(γ (t∗∗
i ))|M +|π(γ (t∗∗
i ))π(γ (ti+1))|M
≤1ε|γ (ti )γ (t∗
i )|M + |γ (t∗
i )γ (t∗∗
i )|M +1ε|γ (t∗∗
i )γ (ti+1)|M
≤1ε
(|γ (ti )γ (t∗
i )|M + |γ (t∗
i )γ (t∗∗
i )|M + |γ (t∗∗
i )γ (ti+1)|M)
=1ε|γ (ti )γ (ti+1)|M .
where the first inequality uses the triangle inequality, the second uses (2-9) and(2-10), and the third uses ε ≤ 1. Hence, noting that π [xy]M is a path in M fromπ(x) to π(y), we have
|π(x)π(y)|M ≤ |π [xy]M |M = lim supN−1∑i=1
|π(γ (ti ))π(γ (ti+1))|M
= lim supN−1∑i=1
|π(γ (ti ))π(γ (ti+1))|M
≤ lim supN−1∑i=1
1ε|γ (ti )γ (ti+1)|M =
1ε|xy|M ,
where the lim sup is taken over all partitions (t1, . . . , tN ) of [0, 1] and over all N .Here the inequality uses (2-8), (2-9), and the previous estimate. �
2.2. Corollaries of the construction. We will now prove Theorems 1.1–1.5 andTheorem 1.9. For the rest of this section, a single choice of warping function φ, asin the construction of the (Alexandrov) extension, will be fixed.
186 JEREMY WONG
Proof of Theorem 1.1(i). From the coarea formula, valid for Alexandrov spaces,
vol(M)=
∫ d(M)
0voln−1(S(x, r; M)) dr
for any x ∈ M , where voln−1 stands for the (n−1)-dimensional Hausdorff measureand S(x, r; M) denotes the metric distance sphere of radius r about x in M . Inparticular, if S(x, r; M)= ∅ then its measure is assigned the value zero. Note thatS(x, r; M)∩ M = ∅ if x is chosen to lie in M and r > d(M). Hence (for example,[Burago et al. 2001, Theorem 10.6.8]),
vol(M)=
∫ d(M)
0voln−1(S(x, r; M)∩ M) dr
≤
∫ d(M)
0voln−1(S(x, r; M)) dr
≤ vol(Sn−1(1))∫ d(M)
0snn−1
min{K −,K −
C }(r) dr,
where
snk(r) :=
sin(
√kr)/
√k k > 0,
r k = 0,sinh(
√|k|r)/
√|k| k < 0
is the generalized sine function; so the volume is bounded above by a constantV (n, K −, λ±, d). �
Proof of Theorem 1.1(ii). Since CM ⊂ M , vol(CM)≤ vol(M)≤ V (n, K −, λ±, d),where the last V is obtained like in Theorem 1.1(i). Since CM = ∂M×φ[0, t0] and∂M×φ{t0} = ε ·
(∂M×φ{0}
),
vol(CM)≥ vol(∂M×φ{t0}) · t0 = εn−1· vol(∂M×φ{0}) · t0.
Combining these inequalities yields
vol(∂M)= vol(∂M×φ{0})≤V (n, K −, λ±, d)
t0 ·εn−1 . �
Remarks on volume. First, even in dimension 2, Theorem 1.1(i) (not requiring λ+)is apparently inaccessible by the Gauss–Bonnet formula, isoperimetric inequalities,and the Heintze–Karcher approach of exponentiating the normal bundle of theboundary, which would work if in addition one had an upper volume bound ofthe boundary itself.
Second, by replacing d with r in Theorem 1.1(i), one also gets a bound for thevolume of a ball B(x, r) in any M ∈ M(n, K −, λ±, d). Neither this nor Theorem1.1(i) itself is sharp in general. It is sometimes more efficient to add a cap or coneto the boundary.
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 187
In contrast to the closed case, there is no volume rigidity for manifolds withsectional curvature and second fundamental form bounds, unless special restric-tions are placed on these values, for example, by assuming the boundary is locallyconvex. In other words, even though it is possible to find a model space for which
vol(B(x, r; M))≤ vol(B(x, r; model space))
holds for every M ∈M(n, K −, λ±, d) and all x ∈ M and 0<r ≤d (we accomplishedthis above), it will not be possible in general to assert that the ball is isometric tothe model space ball when the volume is maximal.
This lack of volume rigidity persists even for small metric balls in the manifold(in particular, those that intersect the boundary) and is related to the observation thatthere appears to be no universal model space or single class of model spaces. Anexample may be constructed by considering the manifolds obtained by removingfrom the unit sphere Sn(1) three disjoint metric balls of the same radius but ofvarious configurations.
Proof of Corollary 1.2. For each Mi , construct Mi using a fixed warping functionindependent of i . By precompactness, the Mi have a Gromov–Hausdorff limit Y .
By Lemma 2.2 the maps πi : Mi → Mi are surjective and uniformly Lipschitz.By Proposition B.1, there is a Lipschitz surjection from Y to X . Lipschitz mapsdo not raise Hausdorff dimension, and so dimH X ≤ dimH Y ≤ n. �
We note as an aside that another proof may be obtained from the volume esti-mate of Theorem 1.1(i). Briefly stated, it suffices to show that the n-dimensionalHausdorff measure voln(X) of X is finite. Let cov denote the covering function,that is, cov(X, r) is the minimal number of balls of radius r needed to cover X .First note that cov is almost-continuous with respect to GH-convergence, in that ifdGH(Mi , X)≤ ηi → 0, then cov(X, r)≤ cov(Mi , r − ηi ).
Then, setting ωn−1 := vol(Sn−1(1)),
voln(X) := limr↓0ωn−1
nrn cov(X, r) by definition
≤ limr↓0 lim supi→∞
ωn−1n
rn cov(Mi , r − ηi )
≤ limr↓0 lim supi→∞
ωn−1n
rn cov(Mi , r)
= lim supi→∞ limr↓0ωn−1
nrn cov(Mi , r) = lim supi→∞ voln(Mi )
≤ V (n, K −, λ±, d) <∞,
where the final “≤” is by Theorem 1.1(i).
Proof of Theorem 1.1(iii). For any metric space X and radius r ≤ R, define thecapacity cap(r, R; X) of X to be the maximum number of disjoint balls of radiusr in X that can be packed in a ball of radius R.
188 JEREMY WONG
Recall that any M ∈ M admits an extension M with i∂(M) ≥ t0. Also, ∂ M =
ε∂M , where ε is the fixed parameter given in the extension procedure. For any0< r ≤ R, it is obvious that
(2-11) cap(r, R; ∂M)= cap(εr, εR; ε∂M)= cap(εr, εR; ∂ M)
since the former amounts to a relabeling of the units of distance.Let exp⊥ denote the normal exponential map of the boundary ∂ M . Observe
that if B1(x1, εr; ∂ M) and B2(x2, εr; ∂ M) are any two disjoint balls in ∂ M andεr ≤ t0/2, then there exist disjoint balls B(εr; M) and B(εr; M) in the cylindricalregions {exp⊥
t (B1) : 0 ≤ t ≤ t0} and {exp⊥t (B2) : 0 ≤ t ≤ t0} of M , respectively,
with radii commensurately bounded below. A similar statement holds for a disjointcollection of balls. Thus if εr ≤ t0/2,
(2-12) cap(εr, εR; ∂ M)≤ cap(εr, d(M); M).
But since curv M ≥ k = k(K −, λ±), relative volume comparison in M implies
(2-13) cap(εr, d(M); M)≤vol(B(d(M); M))
vol(B(εr; M))
≤vol(B(d + 2t0; M))
vol(B(εr; M))
≤vol(B(d + 2t0; Mn
k ))
vol(B(εr; Mnk ))
≤ Nn,k,d(εr),
where Mnk denotes the standard n-dimensional, simply-connected model space of
constant curvature k, and where the quantity Nn,k,d(εr) is independent of M .On the other hand, let γ : [0, R] → ∂M be a diametral minimal segment in
∂M parametrized with unit speed so that R = d(∂M). Suppose r ≤ R. By theminimality of γ , the open metric balls B(γ ( j2r), r; ∂M) for j = 0, 1, . . . , bR/2rc
must be disjoint.Therefore R/2r ≤ cap(r, R; ∂M). Combining this with (2-11)–(2-13) yields
R ≤ 2r Nn,k,d(εr).
In particular, setting r = t0/2ε gives
d(∂M)≤ max{
t02ε,
t0ε
∫ d+2t00 snn−1
k (t)dt∫ t0/20 snn−1
k (t)dt
},
and the latter term yields the upper bound. �
An exponential dependence of the diameter of the boundary upon the diameterof the manifold itself is necessary in general, as the example of a ball in hyper-bolic space shows. One expects that a sharper dimension-free upper bound to the
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 189
intrinsic diameter of the boundary would be (π/ε) sinh(√
|k|(d(M)+ 2t0))/√
|k|,since ∂M =
1ε∂ M , the boundary ∂ M is totally geodesic in M , and M has curvature
bounded below by k = k(K −, λ±) and diameter bounded above by d(M)+ 2t0.However, the optimal estimate (for general values of K −, λ±, and d) seems to beunknown.
Proof of Theorem 1.1(iv). The upper bound on the number of boundary componentsfollows from the Betti number theorem for Alexandrov spaces [Liu and Shen 1994]together with an easy homology argument.
As any M ∈ M(n, K −, λ±, d) is homeomorphic to its extension M , it sufficesto give a bound on the number of components of ∂ M . Consider the double 2M =
M⋃∂ M M . This is again an Alexandrov space of curvature bounded below, since
M has a convex boundary. The Mayer–Vietoris sequence
· · · → H(2M)→ H(∂ M)→ H(M)⊕ H(M)→ · · ·
gives, for p = 0,
rank Hp(∂M)= rank Hp+1(2M)+rank Hp(M)+rank Hp(M) < c(n, K −, λ±, d),
which is finite.Explicitly, from [Liu and Shen 1994], one has the superexponential bound
n∑p=0
rank Hp(M)≤ c(n, K −, λ±, d)= C25n+n+2
n (√
|k|d + 1)ne3n√
|k|d
for the total Betti number, where Cn = 12n2+3n+1 and k = k(K −, K −
C ) is the lowercurvature bound of M . �
We remark that the number of boundary components will be at most two ifthere is enough combined positivity of curvature in the interior and convexity ofthe boundary.
We next consider precompactness of the class M(n, K −, λ±, d).
Proof of Theorem 1.3. Take any M ∈ M(n, K −, λ±, d). Let B(x, r) denote aball in an extension M as above, with center x and radius r . We claim that for anyx, y ∈ M and r >0 that (i) B(x, r)∩ B(y, r)=∅ implies B(x, r)∩B(y, r)=∅ andconversely (ii) B(x, r)∩ B(y, r)= ∅ is implied by B(x, f (2r))∩ B(y, f (2r))= ∅for some function f .
Part (i) is obvious, since B(x, r) ⊇ B(x, r) and B(y, r) ⊇ B(y, r). For theconverse (ii), suppose z ∈ B(x, r)∩ B(y, r). Then |xz|M , |yz|M < r . Consider the
190 JEREMY WONG
point z′:= π(z), where π : M → M is the projection map defined in (2-7). Then
|xz′|M ≤ |xz|M + |zz′
|M
≤ |xz|M + |xz|M by choice of z′
< r + r since |xz|M < r .
Equivalently, z′∈ B(x, 2r; M).
We now assert that z′∈ B(x, f (2r); M) for some function f . (A symmetric
argument with y in place of x will give z′∈ B(y, f (2r); M), so that, as desired,
B(x, f (2r); M)∩ B(y, f (2r); M) 6= ∅.) For this, it suffices to demonstrate that|xz′
|M ≤ r implies |xz′|M ≤ f (r). But this is immediate from property (iii) on page
184 (as proved in Lemma 2.2), whereby we may take f (r)= r/ε. This shows thatthe claim above holds.
For any metric space X , recall the definition of capacity (cap) from page 187.To show that M(n, K −, λ±, d) is precompact, we will show that cap(r, R; ·) isbounded on the class M(n, K −, λ±, d). It suffices to demonstrate that cap(r, d; ·)
is so bounded for all sufficiently small r . Suppose M ∈ M(n, K −, λ±, d).Set d := d + 2t0, and recall that d(M)≤ d . In particular, B(d; M)= M . Then
cap(r, d; M)= cap(r, d; M)
:= maximum # disjoint r -balls B(r; M) in a d-ball B(d; M)
≤ maximum # disjoint ε2r -balls B( ε2r; M)
with centers in M in a d-ball B(d; M) by the above claim
≤ maximum # disjoint ε2r -balls B( ε2r; M) in M
=: cap( ε2r, d; M),
and the latter is bounded by relative volume comparison in M , in terms of only n,K −
= K −(K −, λ±), λ−= 0, and d , hence by only n, K −, λ±, and d. �
Proof of Theorem 1.4. Any M ∈ M(n, K −, λ±, vol ≥ v > 0, d) is homeomorphicto its extension M ∈ M(n, curv ≥ k(K −, λ±), vol ≥ v, d), and the latter classcontains only finitely many homeomorphism types, by the topological stabilitytheorem of [Perelman 1992]. �
Proof of Theorem 1.5. Existence of the limit of the Mi ’s follows (after possi-bly choosing a subsequence) from precompactness, since the Mi are Alexandrovspaces satisfying d(Mi )≤ d + 2t0 <∞.
For each i , one has a map πi : Mi → Mi , as defined in (2-7). Lemma 2.2 andthe first part of Proposition B.1 imply that πi → π for some map π : limGH Mi →
limGH Mi .
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 191
The inclusions ιi : Mi → Mi (with Lipschitz constant equal to 1) subconvergeto a map ι : limGH Mi → limGH Mi , again by the first part of Proposition B.1.
One can write any point x ∈ CMi = ∂Mi×φ[0, t0] in coordinates as x = (x ′, r),where x ′
∈ ∂Mi and r ∈ [0, t0].Define a map Hi,t : Mi → Mi , for any time 0 ≤ t ≤ 1, by
Hi,t(x)=
{x if x ∈ Mi ,(x ′, tr) if x = (x ′, r) ∈ CMi .
Then Hi,0(x)= ιi ◦πi (x) and Hi,1(x)= idMi(x)= x .
Let Hi (x, t) = Hi,t(x). For any x, y ∈ Mi and times 0 ≤ s, t ≤ 1, we will in amoment show
(2-14) |Hi (x, t)Hi (y, s)|Mi ≤ L|(x, t)(y, s)|Mi ×I
for some constant L = L(ε, t0) <∞. Here, the metric on the space-time Mi × I istaken to be the direct product metric.
Given inequality (2-14), the proof of Theorem 1.5 can be finished as follows.By Proposition B.2, the homotopies Hi,t from ιi ◦ πi to idMi give rise to a limithomotopy ι ◦π ' idlimGH Mi . On the other hand, for all i , πi ◦ ιi = idMi on Mi , soπ ◦ ι= idlimGH Mi . Therefore limGH Mi ' limGH Mi .
It only remains to prove inequality (2-14). To accomplish this, there are onlythree cases to consider:
Case: x, y ∈ Mi .
(2-15) |Hi (x, t)Hi (y, s)|Mi = |xy|Mi by definition of Hi
≤ |(x, t)(y, s)|Mi ×I for any s, t ∈ I .
Case: x, y ∈ CMi . This case requires a preparatory inequality, Lemma 2.3 below,whose proof is given shortly after we otherwise finish proving Theorem 1.5.
Write x = (x ′, u) and y = (y′, v). Then Lemma 2.3 shows that
(2-16) |(x ′, tu)(y′, tu)|Mi ≤1ε|(x ′, u)(y′, u)|Mi
Proceeding, observe that
(2-17) |uv|I ≤ |(x ′, u)(y′, v)|Mi = |xy|Mi
We will also use the algebraic inequality that
(2-18) |tu − sv| ≤√
2 max{1, t0} · (|u − v|2 + |t − s|2)1/2
192 JEREMY WONG
whenever 0 ≤ t ≤ 1 and 0 ≤ v ≤ t0. To see this, write tu −sv= t (u −v)+v(t −s),so that
|tu − sv| ≤ max{t}|u − v| + max{v}|t − s|
≤ max{max t,max v}(|u − v| + |t − s|)
≤ max{max t,max v}√
2(|u − v|2 + |t − s|2)1/2.
Hence
|Hi (x, t)Hi (y, s)|Mi = |(x ′, tu)(y′, sv)|Mi by definition of Hi
≤ |(x ′, tu)(y′, tu)|Mi + |(y′, tu)(y′, sv)|Mi by the triangle inequality
≤1ε|(x ′, u)(y′, u)|Mi + |(y′, tu)(y′, sv)|Mi by (2-16)
=1ε|(x ′, u)(y′, u)|Mi + |tu − sv|I
≤1ε|(x ′, u)(y′, u)|Mi + max{1, t0}
√2(|uv|2I + |ts|2I )
1/2 by estimate (2-18) above
≤1ε(|(x ′, u)(y′, v)|Mi + |(y′, v)(y′, u)|Mi )+ max{1, t0}
√2(|uv|2I + |ts|2I )
1/2
by the triangle inequality
=1ε(|xy|Mi + |uv|I )+ max{1, t0}
√2(|uv|2I + |ts|2I )
1/2
≤1ε(|xy|Mi + |xy|Mi )+ max{1, t0}
√2(|xy|
2Mi
+ |ts|2I )1/2
by the observation (2-17) above=
2ε|xy|Mi + max{1, t0}
√2|(x, t)(y, s)|Mi ×I
≤2ε|(x, t)(y, s)|Mi ×I + max{1, t0}
√2|(x, t)(y, s)|Mi ×I
=( 2ε+ max{1, t0}
√2)|(x, t)(y, s)|Mi ×I
Case: x ∈ Mi , y ∈ CMi . Choose z ∈ ∂Mi and r ∈ I such that (z, r) belongs to aminimal geodesic [(x, t)(y, s)]Mi ×I . This is possible since [xy]Mi must cross ∂Mi ,which implies that [(x, t)(y, s)]Mi ×I must cross the subset ∂Mi × I ⊂ Mi × I .
|Hi (x, t)Hi (y, s)|Mi ≤ |Hi (x, t)Hi (z, r)|Mi + |Hi (z, r)Hi (y, s)|Mi
by the triangle inequality≤ |(x, t)(z, r)|Mi ×I + |Hi (z, r)Hi (y, s)|Mi
by (2-15) of the first case, since x, z ∈ Mi ,
≤ |(x, t)(z, r)|Mi ×I +( 2ε+ max{1, t0}
√2)|(z, r)(y, s)|Mi ×I
by the previous case’s result, since z, y ∈ CMi ,
≤ max{1,( 2ε+ max{1, t0}
√2)}(
|(x, t)(z, r)|Mi ×I + |(z, r)(y, s)|Mi ×I)
=( 2ε+ max{1, t0}
√2)|(x, t)(y, s)|Mi ×I by choice of z and r .
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 193
Considering the results of these three cases together, this proves inequality(2-14) and hence Theorem 1.5. �
Lemma 2.3. In the setting as above, |(x ′, tu)(y′, tu)|Mi ≤1ε|(x ′, u)(y′, u)|Mi .
Proof. We can prove this using a segment subdivision method virtually identicalto that in Lemma 2.2.
Analogously to the projection-type map defined in (2-7), define a map
πtu : Mi → Mi ∪ (∂Mi×φ[0, tu])
by
πtu(w)=
{w if w ∈ Mi ∪ (∂Mi×φ[0, tu]),
orthogonal projection of w onto ∂Mi×φ{tu} if w ∈ ∂Mi×φ[tu, t0].
(So in particular, π0 is just the map π from Lemma 2.2.)Now the proof may be repeated verbatim from Lemma 2.2, if one substitutes
Mi ∪ (∂Mi×φ[0, tu]) for Mi , ∂Mi×φ[tu, t0] for CMi , and πtu for π. �
Now we generalize several of the results above to Ricci curvature.
Proof of Theorem 1.9. (i) Fix a constant t0 as in Lemma 1.7. For any M ∈
M(n, r−, λ±, d), consider its smoothed extension Mδ, which results from applyingthe Ricci version of Lemma 1.7 together with Theorem 1.8. We may assume thatδ has been chosen so that δ < t0. We will restrict δ further in a moment. Letr−
1 := min{r−, r−
CM} be the minimum of the lower Ricci curvature bound of M and
the lower Ricci curvature bound r−
CM= r−
CM(r−, λ±) of CM . Then2 ric(Mδ)≥ r−
1 .Since the smoothing on the δ-neighborhood of ∂M is effected through linear
interpolations of the metrics of M and CM written in the form g = dt2+gt , where
t ∈ (−δ, δ) denotes the signed distance to ∂M , and the metric of M⋃∂MCM is
unchanged off the δ-neighborhood of ∂M , it follows that
MδGH−→M
⋃∂MCM as δ → 0.(2-19)
Thusd(Mδ)≤ d(M
⋃∂MCM)+ τ(δ)≤ d + 2t0 + τ(δ),
where τ is a function satisfying τ(δ) → 0 as δ → 0. We can assume δ is chosensufficiently small so that τ(δ)≤ 1, say.
2Technically, Theorem 1.8 was stated only for strict inequality of the Ricci curvature tensor, butif ric(M)≥ r− and ric(CM )≥ r−
CMthen ric(M) > r−
−η and ric(CM ) > r−CM
− η for all η > 0. So,provided that δ is sufficiently small relative to M and CM , ric(Mδ) >min{r−
− η, r−CM
− η} for allη > 0 by Theorem 1.8, which implies ric(Mδ)≥ r−
1 .
194 JEREMY WONG
Since Mδ has totally geodesic boundary by construction, absolute volume com-parison applies to Mδ, and
vol(Mδ)≤ V (n, r−
1 , d + 2t0 + 1)= V (n, r−, λ±, d).
Since M is not isometrically embedded in Mδ, it cannot be asserted directlythat vol(M) ≤ vol(Mδ). However, (2-19) implies vol(Mδ)→ vol(M
⋃∂MCM) by
volume continuity [Cheeger and Colding 1997, Theorem 5.9]. Since M is isomet-rically embedded in M
⋃∂MCM , it follows that
vol(M)≤ vol(M⋃∂MCM)≤ vol(Mδ)+ τ(δ)≤ V (n, r−, λ±, d)
for some upper bound V (n, r−, λ±, d).
(ii) The proof is similar to that of Theorem 1.1(iv), except that here instead ofthe Betti number for Alexandrov spaces, we can use that closed n-dimensionalmanifolds having lower Ricci curvature bound and upper diameter bound havebounded first Betti number [Gromov 1999, 5.20, 5.21].
(iii) The proof is exactly the same as in Theorem 1.1(iii), with Mδ in place of Mand i∂(M)≥ t0 replaced by i∂(Mδ)≥ t0 − δ.
(iv) The proof is the same as in Theorem 1.3, although the very last line there,which invoked relative volume comparison in M , needs here further justification.Given that M and CM only have a lower Ricci curvature bound, it is still truethat relative volume comparison holds for the glued space M = M
⋃∂MCM , by
[Cheeger and Colding 1997, Theorem 5.9], since the convergence (2-19) of Mδ
to M is noncollapsing in the sense that vol(B(x, 1; Mδ)) ≥ v > 0 for some v andall x and δ > 0. The model space for the relative volume comparison here is thesimply-connected n-dimensional space of constant curvature r−
1 . �
2.3. Questions. It would be interesting to see if the upper bound λ+ could beeliminated from Theorems 1.1–1.5 and Theorem 1.9. The only place where thisbound was used was in invoking the Gauss equations to produce a lower curvaturebound for the extension.
It also remains to be seen for which of the results can the lower bound λ− onthe second fundamental form be replaced by a lower bound on the mean curvature,or perhaps by a bound on integral of mean curvature.
In this direction of considering weaker curvature bounds, we remark that aprecompactness and convergence theorem has recently been proved in [Ander-son et al. 2004], in the noncollapsing regime for which one has all of the in-jectivity radius bounds for iint(M), inj(∂M), and i∂(M). They considered theclass M(n, r0, i0, H0, d), consisting of manifolds with boundary (M, ∂M) suchthat dim M = n, ‖ ric(M)‖L∞(M) ≤ r0, ‖ ric(∂M)‖L∞(∂M) ≤ r0, iint(M) ≥ i0,
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 195
inj(∂M) ≥ i0, i∂ ≥ i0, ‖H‖Lip(∂M) ≤ H0, and d(M) ≤ d , and showed that itis precompact in the Cr topology for each r < 2. The main idea of their proofinvolved boundary harmonic coordinates and boundary harmonic radius.
Theorem 1.4 suggests that there is an underlying stability accounting for thefiniteness. Namely, if N ∈ M = M(n, K −, λ±, vol>v > 0, d) then does there existan ε > 0 such that any M ∈ M with dGH(M, N ) < ε is homeomorphic to N? Thiscould be answered if it could be shown that Gromov–Hausdorff closeness of twomanifolds implied their (Alexandrov) extensions were correspondingly close.
Appendix A. Injectivity radii: definitions
In a manifold with boundary, the usual Riemannian exponential map is not well de-fined because geodesics may bifurcate. Here we give two definitions of injectivityradii which are used in this paper.
Definition. For a Riemannian manifold with boundary (M, ∂M), and p ∈ M , de-fine iint(p) to be the supremum over all r > 0 such that any unit-speed geodesicγ : [0, tγ ]→ M issuing from p is distance minimizing up to the distance min(tγ , r),where tγ is the first time γ intersects ∂M (so tγ = ∞ if γ ∩ ∂M = ∅). Defineiint(M)= infp∈M{iint(p)}, the interior injectivity radius.
For a closed manifold M , or a manifold with locally convex boundary, iint(M)coincides with the usual notion of injectivity radius, often denoted inj(M), definedvia the exponential map.
Definition. For a Riemannian manifold with boundary (M, ∂M), and p ∈ ∂M ,define i∂(p) to be the supremum over all r > 0 such that any minimizing geodesicγ issuing from p normally to ∂M uniquely minimizes distance to ∂M up to distancer (that is, γ (0) = p and d(γ (r), ∂M) = r ). Define i∂(M) = infp∈∂M{i∂(p)}, theboundary injectivity radius of (M, ∂M).
So, if i∂(M) ≥ i0, then M admits an inward tubular neighborhood of radiusat least i0. One has the Klingenberg-type estimate i∂(M) = min{Foc(∂M), L/2},where Foc(∂M) is the minimum focal distance for the normal exponential mapof the boundary and L represents the shortest length of a segment meeting ∂M atright angles at both its endpoints. It is known that
Foc(∂M)≥1
√K +
arctan(√
K +
λ+
)if KM ≤ K + and II∂M ≤ λ+.
An example of these definitions is M = (Rn\ Bn(r), gstd), the Euclidean space
with a ball of radius r removed. It has iint(M)= ∞ and i∂(M)= ∞.
196 JEREMY WONG
Another is M = (Sn(1)\ Bn(r), gstd), the standard sphere of radius 1 with a ballof radius r removed, which, for 0< r < π , has
iint(M)=
{π for r ≤ π/2,∞ for r > π/2,
and i∂(M)= π − r.
Appendix B. Gromov–Hausdorff convergence
This section gives background on Gromov–Hausdorff convergence. In particular,it details two functorial properties of maps: one for surjective, Lipschitz mapsand another for Lipschitz homotopy equivalences. These are used for the proofsof Corollary 1.2 and Theorem 1.5. Additional background on Gromov–Hausdorffconvergence may be found in [Fukaya 1990; Petersen 1993].
Let Z be a metric space. The Hausdorff distance d ZH (X, Y ) between two subsets
X, Y ⊆ Z is defined to be d ZH (X, Y ) := inf{ε > 0 : B(X, ε) ⊇ Y, B(Y, ε) ⊇ X},
where B(X, ε)= {z ∈ Z : d(z, X) < ε} denotes the metric Z -ball about X of radiusε.
Definition. The Gromov–Hausdorff distance between two metric spaces X and Yis dGH(X, Y )= inf{ d Z
H (iX (X), iY (Y ))}, where the infimum is taken over all metricspaces Z and all distance-preserving embeddings iX : X ↪→ Z and iY : Y ↪→ Z .
We say a sequence of metric spaces X i converges to X , and write X iGH−→X , if
dGH(X, X i )→ 0 as i → ∞. In practice one usually uses the following formulationto verify that a convergence occurs.
Definition. An ε-Hausdorff approximation ψ : X → Y is a (not necessarily contin-uous) map such that ψ(X) is an ε-net in Y , that is, B(ψ(X), ε; Y )= Y , and ψ is anε-almost isometry, that is, |dY (ψ(x1), ψ(x2))− dX (x1, x2)| ≤ ε for all x1, x2 ∈ X .
One has the notion of convergence of points.
Definition. If X iGH−→X via εi -Hausdorff approximations ψi : X i → X , we say
points xi ∈ X i converge to a point x ∈ X (and write xi 7→ x) if d(ψi (xi ), x)→ 0.
This permits one to define convergence of maps.
Definition. If fi : X i −→ Yi are maps, X iGH−→X and Yi
GH−→Y , then the fi converge
to a map f : X → Y if fi (xi ) 7→ f (x) whenever X i 3 xi 7→ x ∈ X ; for this wewrite fi → f .
The following proposition is a slight modification of [Gromov 1999, Section3.11 1
2 +, exercise (c) on p. 78].
Proposition B.1. Let X i and Yi be metric spaces with X iGH−→X and Yi
GH−→Y , with
X and Y compact. Suppose that for all i there exist L-Lipschitz maps fi : Yi → X i .Then there exists an L-Lipschitz map f : Y → X. If the fi are also surjective, thenso is the limit map f .
AN EXTENSION PROCEDURE FOR MANIFOLDS WITH BOUNDARY 197
For the proof, see [Wong 2006].
Proposition B.2. Suppose X jGH−→X and Y j
GH−→Y for complete metric spaces X j
and Y j , with X and Y compact. Suppose that for each j there exist continuousmaps f j : X j → Y j , g j : Y j → X j , f : X → Y , and g : Y → X such that f j → fand g j → g.
Suppose that for each j , there exist maps H j : X j × I → X j such that
H j (x, 0)= g j ◦ f j (x), H j (x, 1)= idX j (x)= x,
and H j (x, t) is globally Lipschitz in x, t , uniformly in j (where X j × I is equippedwith the direct product metric). Suppose also there exist maps H j : Y j × I → Y j
for each j such that
H j (x, 0)= f j ◦ g j (x), H j (x, 1)= idY j (x)= x,
and H j (x, t) is globally Lipschitz in x, t , uniformly in j (where Y j × I equippedwith the direct product metric). Then X and Y are homotopy equivalent (via aLipschitz homotopy equivalence).
The theorem is illustrated by this diagram:
X j
GH��
f j
'
//Y j
GH��
g joo
Xf //
Yg
∴ 'oo
Proof. By assumption, there exists L <∞ such that
|H j (x, t)H j (y, s)|X j ≤ L|(x, t)(y, s)|X j ×I(B-1)
for all x, y ∈ X j and all 0 ≤ s, t ≤ 1; in particular, the H j are equicontinuous.Since X is compact and hence bounded, d(X j ) ≤ d <∞ for all sufficiently largej , whence the H j are also uniformly bounded.
Note that X j × I GH−→X × I since X j
GH−→X .
By Arzela–Ascoli, H j subconverge to a map H : X × I → X satisfying
(B-2) |H(x, t)H(y, s)|X ≤ L|(x, t)(y, s)|X×I
for all x, y ∈ X and all 0 ≤ s, t ≤ 1, by the first part of Proposition B.1. Inparticular, H is jointly continuous with respect to x, t . Also, since g j ◦ f j → g ◦ fand idX j → idX , we have H(x, 0)= g◦ f (x) and H(x, 1)= idX (x)= x . ThereforeH is a homotopy from g ◦ f to idX , that is, g ◦ f ' idX .
A symmetric argument, with H j in place of H j , yields a homotopy H betweenf ◦ g and idY , that is, f ◦ g ' idY . Therefore X ' Y . �
198 JEREMY WONG
If the limiting homotopies are not uniformly (in j) Lipschitz, the limit spacesneed not be homotopy equivalent. For example, take X j (for all j) identically to bea given point {x} in the unit sphere S2(1), and let Y j := S2(1)\ B2(−x, 1/j), where−x denotes the antipode to x . Let H j be the obvious deformation retraction of Y j
to X j . Although Y j ' X j for each j , the limit space Y = S2(1) is not homotopyequivalent to X = {x}.
Acknowledgment
The author thanks Professor Stephanie Alexander for helpful discussions and sug-gestions.
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Received May 2, 2007.
JEREMY WONG
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF TORONTO
TORONTO, ON M5S 2E4CANADA
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PACIFIC JOURNAL OF MATHEMATICS
Volume 235 No. 1 March 2008
Powers of theta functions 1HENG HUAT CHAN AND SHAUN COOPER
Optimal oscillation criteria for first order difference equations with delayargument 15
GEORGE E. CHATZARAKIS, ROMAN KOPLATADZE AND IOANNIS P.STAVROULAKIS
Generalized handlebody sets and non-Haken 3-manifolds 35JESSE EDWARD JOHNSON AND TERK PATEL
Left-symmetric superalgebraic structures on the super-Virasoro algebras 43XIAOLI KONG AND CHENGMING BAI
Deformations of nearly Kahler structures 57ANDREI MOROIANU, PAUL-ANDI NAGY AND UWE SEMMELMANN
Infinite-dimensional sandwich pairs 73MARTIN SCHECHTER
On algebraically integrable outer billiards 89SERGE TABACHNIKOV
Bornological quantum groups 93CHRISTIAN VOIGT
Deformations of the gyroid and lidinoid minimal surfaces 137ADAM G. WEYHAUPT
An extension procedure for manifolds with boundary 173JEREMY WONG
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