Complete, Embedded, Minimal n-Dimensional Submanifolds in Cn CLAUDIO AREZZO Università di Parma AND...

Complete, Embedded, Minimal n-DimensionalSubmanifolds in C

n

CLAUDIO AREZZOUniversità di Parma

AND

FRANK PACARDUniversité Paris XII

1 Introduction

A classical problem in the theory of minimal submanifolds of Euclidean spacesis to understand whether a minimal submanifold exists with a prescribed behaviorat infinity, or to determine from the asymptotes the geometry of the whole sub-manifold. Beyond the intrinsic interest of these questions, they are also of crucialimportance when studying the possible singularities of minimal submanifolds ingeneral Riemannian manifolds. For minimal surfaces, i.e., two-dimensional sub-manifolds, the standard tool to solve these problems is given by the Weierstrassrepresentation formula, which relates the geometry of the minimal surface to com-plex analytic properties of holomorphic 1-forms on Riemann surfaces. Recentlygluing techniques have been developed that provide an abundant number of newexamples of minimal hypersurfaces in Euclidean space.

For higher-dimensional minimal submanifolds such a link clearly disappearsand no complex analysis can be put into play. Although gluing techniques havebeen extensively used in the study of minimal hypersurfaces, they have not beenadapted to handle higher codimensional submanifolds. The aim of this paper isto use a gluing technique for minimal submanifolds to make a step toward under-standing these questions in higher codimension. More precisely, we will restrictourselves to the case of real n-dimensional submanifolds of C

n . There are twomain reasons for doing so. The first one comes from the fact that when tryingto desingularize the intersection, for example, of a pair of n-planes that give thedesired asymptotic behavior, one needs a model of a minimal submanifold withthis behavior at infinity to rescale and to glue into the pair of planes where aneighborhood of the intersection is removed. This local model needs to be suf-ficiently simple to allow a detailed study of the linearized mean curvature operator.In our situation this is provided by a generalization of an area-minimizing subman-ifold found by Lawlor in [12], while in more general cases such an example is notknown. The second reason is that, among minimal n-submanifolds of C

n , there is a

Communications on Pure and Applied Mathematics, Vol. LVI, 0283–0327 (2003)c© 2003 Wiley Periodicals, Inc.

284 C. AREZZO AND F. PACARD

special family, namely, the special Lagrangian ones, which are of great importancein a variety of geometric and physical problems (see, for example, [9, 11, 22]).

To better describe our result, let us first observe that, for minimal surfaces inC

2, graphs of meromorphic (or antimeromorphic) functions are enough to producemany interesting surfaces. For example, given z1, . . . , zk ∈ C all distinct andα1, . . . , αk ∈ C, the surface that is the graph of

(1.1) z ∈ C \ {z1, . . . , zk} −→ α1(z − z1)−1 + · · · + αk(z − zk)

−1 ∈ C2 ,

is a complete embedded minimal surface with k + 1 ends.When k = 1, the hyperbola is parametrized by

z ∈ C \ {0} −→ (z, z−1) ∈ C2 .

This surface has a natural generalization in any dimension. In fact, the n-dimen-sional hyperbola HI of C

n is defined by(0,

π

n

)× S

n−1 � (s,�) −→ eis

(sin(ns))1/n� ∈ C

n .

For any n ≥ 2, HI is a complete special Lagrangian (and therefore minimal) sub-manifold of C

n with two ends; we refer to [8, 9, 12] for further properties of thesesubmanifolds.

Observe that, for any R ∈ O(n), we can define HR to be the n-dimensionalsubmanifold of C

n that is parametrized by(0,

π

n

)× S

n−1 � (s,�) −→ 1

(sin(ns))1/n(cos s� + i sin sR(�)) ∈ C

n .

This is still a minimal submanifold since it is the image of HI by an isome-try. However, given the complex structure in R

2n induced by the identificationof (x, y) ∈ R

2n with x + iy ∈ Cn , it fails to be Lagrangian except when R2 = I .

In this paper, we prove the existence of minimal embedded n-dimensional sub-manifolds that have k + 1 ends and that, in some sense, can be understood as thehigh-dimensional analogues of the surfaces described by (1.1). To make thingsprecise, let us consider the “horizontal” n-plane

�0 := {x ∈ C

n : x ∈ Rn} ,

and, for all j = 1, . . . , k, further consider the n-planes

�j :={(

x j + cosπ

nx + i sin

π

nRj x

)∈ C

n : x ∈ Rn}

,

where x1, . . . , xk ∈ Rn and R1, . . . ,Rk ∈ O(n) are fixed. We further assume that

points x j are all distinct.In order to highlight a geometric property of these planes, let us recall the notion

of “characterizing angles” between two n-planes � and �′ in Cn: Among all pairs

of unit vectors v ∈ � and w ∈ �′, choose v1 and w1 to minimize the anglebetween them. Next choose v2 ∈ � and w2 ∈ �′ with v2 orthogonal to v1 and

MINIMAL N -DIMENSIONAL SUBMANIFOLDS IN CN 285

w2 orthogonal to w1, again minimizing the angle between such pairs. Repeat thisprocedure n − 1 times, and finally define vn and wn to get an oriented basis of �

and �′. The i th characterizing angle between � and �′ is then the angle betweenvi and wi . For example, it is easy to check that the angles between the planes �0and �j are all equal to π

n [9].Granted x1, . . . , xk ∈ R

n and R1, . . . ,Rk ∈ O(n), we define for all j �= j ′ thevectors

ξj, j ′ := x j − x j ′

|x j − x j ′ |and the k × k matrix � := (γj j ′)j, j ′ with diagonal entries γj j = 0, for all j =1, . . . , k, and off-diagonal entries

γj j ′ := 1

|x j − x j ′ |n∫

Sn−1

(Rj� · Rj ′� − n(� · Rjξj, j ′)(� · Rj ′ξj, j ′))dθ

for all j �= j ′.Finally, given A0 ∈ Mn(R), we define a vector := (λ1, . . . , λk) where, for

all j = 1, . . . , k,

λj := −∫

Sn−1

A0� · Rj� dθ .

In order for the desingularization procedure to be possible, we make the fol-lowing assumptions:

(H1) ∀ j �= j ′, ξj j ′ /∈ Im(I − R−1j ′ Rj ).

(H2) The k × k matrix � is invertible.(H3) ∈ �(0,+∞)k .

We will comment on these assumptions in the next section. We can now state themain result of the paper.

THEOREM 1.1 Assume that n ≥ 3, and that (H1), (H2), and (H3) are fulfilled.Then, the set of n-planes �0, . . . ,�k can be desingularized to produce a family ofcomplete, embedded, minimal n-dimensional submanifolds of C

n that have k + 1planar ends. Each member of this family is of the topological type of an n-spherewith k + 1 punctures and has finite total curvature.

More precisely, there exists ε0 and, for all ε ∈ (0, ε0), there exists �ε, an n-dimensional submanifold of C

n such that the following hold:

(i) For all ε ∈ (0, ε0), �ε is embedded and minimal.

(ii) For all ε ∈ (0, ε0), �ε has k + 1 ends, which, up to (small) translations,are given by �j for j = 1, . . . , k and {x + iεA0x : x ∈ R

n}.(iii) As ε tends to 0, �ε converges, on compact sets of C

n and away from thepoints {x1, . . . , xk}, to

⋃kj=0 �j in the C∞ topology.


(iv) Define (α1, . . . , αk) := �−1. Then, as ε tends to 0 and for all j =1, . . . , k, the family of rescaled submanifolds α−1

j ε−1/n(�ε − x j ) converges oncompact sets to HRj in the C∞ topology.

If (H1), (H2), and (H3) are not fulfilled, it is still possible to desingularize⋃kj=0 �j . However, we then obtain minimal submanifolds that are not embedded.

More generally, not every configuration of planes and points can be obtained aslimit of families of immersed minimal submanifolds. For example, M. Ross [18]proved that a 2-sphere with two punctures minimally immersed in R

4 with twosimple planar ends has to be holomorphic, thus imposing a severe restriction onthe planar ends.

A number of questions naturally arise from the above result. In the first place,it would be interesting to know whether conditions (H1), (H2), and (H3) are alsonecessary for such a desingularization to exist. It is easy to check that (H1) is nec-essary to guarantee embeddedness of the submanifold, and we believe that (H2)and (H3) are also necessary conditions, at least when restricting the topologicaltype of the minimal submanifolds. If this turns out to be the case, since there arespecial Lagrangian configurations of planes that do not satisfy these conditions, itwould mean that they cannot be obtained as limits of special Lagrangian submani-folds, thus helping to understand the possible degenerations of these manifolds.

Not unrelated is the problem of determining whether one can desingularize aconfiguration of special Lagrangian planes through special Lagrangian submani-folds, possibly leaving free the phase of the calibration to change as in [19]. In thecase of two planes, Lawlor’s examples obviously answer the question. A. Butscher[3] has recently generalized this to the case of the intersection of two special La-grangian submanifolds with boundary in a general Calabi-Yau manifold. Gluingtechniques applied to special Lagrangian submanifolds of Calabi-Yau threefoldshave also been used by S. Salur in [20, 21]. Finally, we would like to mentionthe recent work of J. Isenberg, R. Mazzeo, and D. Pollack [10], where a technicallinear analysis similar to ours is developed.

The paper is structured as follows: Section 2 provides examples of configura-tions of planes satisfying the assumption of Theorem 1.1. Sections 3 and 4 collectclassical tools from Riemannian geometry useful to studying the linearized meancurvature operator (Jacobi operator) of a minimal submanifold. Sections 5 and 6contain a detailed description of the special Lagrangian hyperbola, which plays thepart of the local model for the desingularization.

In Section 7 the Jacobi operator LH of the hyperbola is computed. The key pointis to find an expression for LH (and of the asymptotic operator L0 approachinginfinity on the hyperbola) in terms of classical operators on the sphere. In thisrespect it is particularly useful to identify the normal vector fields to the hyperbolawith sections of bundles of 0- and 1-forms over the sphere. In Section 8, we collectthe needed results about the spectral properties of the operators on the spheresarising from these Jacobi operators. Next we study the space of solutions of the


problem LHU = 0 on the hyperbola. In Section 9 we study the exponential growth(i.e., the indicial roots) of the solutions, while in Section 10 we find, by geometricalmeans, explicit expressions for families of Jacobi fields on the hyperbola.

Having now a good understanding of the kernel of the operator LH , we turn toits injectivity and surjectivity properties. Proposition 11.1 gives the desired injec-tivity for L0, while in the next two propositions we show that the same result holdsfor LH under the assumption that the function U stays orthogonal to the constantsof S

n−1 (corresponding to normal vector fields via the mentioned identification de-scribed in Section 8) in the L2 sense.

Surjectivity of LH (and the existence of the Green’s function) in a suitableweighted Hölder space is the theme of Section 12. Propositions 12.2 and 12.3,again under the same orthogonality condition as before, provide a right inverse forLH , while Proposition 12.4 shows that no assumptions are required to estimate thenorm of the Poisson operator associated to L0 (a fact that will be crucial to applyinga contraction argument at the end of the proof).

All these properties ultimately serve to parametrize all minimal submanifoldsnear the hyperbola. Up to now we looked only at the linear problem. In Section 13we compute the nonlinear term for the mean curvature of a graph over the hyper-bola, and we estimate Hölder norms of the nonlinear terms. In Section 14 we givean explicit description of all minimal submanifolds that can be written as normalgraphs over (pieces of) the hyperbola. In Sections 15, 16, and 17 an analogue anal-ysis is done to parametrize all minimal submanifolds close to the n-planes twistedby the Rj ’s.

The proof of Theorem 1.1 is completed in Section 18 by comparing the bound-aries of these families of minimal submanifolds and showing that, under assump-tions (H1), (H2), and (H3), they do match; that is, it is possible to glue them to formthe sought families of minimal submanifolds with prescribed asymptotic behavior.

2 Comments and Examples

In this section, we state our assumptions and give examples of sets of pointsx1, . . . , xk ∈ R

n , orthogonal transformations R1, . . . ,Rk ∈ O(n), and matricesA0 ∈ Mn(R) for which (H1), (H2), and (H3) hold. To simplify the discussion, letus define

S := {(x1, . . . , xk) ∈ R

n × · · · × Rn : x j �= x j ′ if j �= j ′}

and := O(n) × · · · × O(n) .

2.1 Comments1. Observe that

Im(I − R−1

j ′ Rj) = Im

(I − R−1

j Rj ′),


so that condition (H1) is symmetric in j and j ′. Now, as may easily be checked,this assumption guarantees that

∀ j �= j ′ �j ∩ �j ′ = ∅ ,

while �0 ∩ �j = {x j } for all j ≥ 0.

2. Obviously, the set of ((x j )j , (Rj )j ) ∈ S × for which (H1) is fulfilledis an open set that is not equal to S × . Indeed, if I − R−1

j ′ Rj is invertible forsome j �= j ′, then (H1) does not hold. Similarly, the set of ((x j )j , (Rj )j ) ∈ S ×

such that (H2) holds is an open set that is not equal to S × . For example, whenall Rj are equal, then � ≡ 0 and hence (H2) does not hold. Finally, the set of((x j )j , (Rj )j ,A0) ∈ S × × Mn(R) for which (H3) is fulfilled is also an open set.

3. In Mn(R) we define the relation A ∼ A′ if and only if there exists R,R′ ∈O(n) such that

A = RA′R′ .Now, if A0 ∼ A′

0 and if (H3) is fulfilled for A0 and (Rj )j ∈ , then (H3) isfulfilled for A′

0 and (RRjR′)j ∈ . Hence, the set of A ∈ Mn(R) such that (H3)is fulfilled for some (Rj )j ∈ only depends on the coset of A in Mn(R)/ ∼.

2.2 ExamplesWhen k = 2, we give some examples for which (H1), (H2), and (H3) are

fulfilled. Without loss of generality, we can assume that x1 = −x2 := e, with‖e‖ = 1. Let us assume that both orthogonal transformations Ri leave e un-changed; that is,

R1e = R2e = e .

Thereforee /∈ Im

(I − R−1

1 R2),

and in particular, (H1) is always fulfilled for such a choice of R j .Next, we would like to compute

γ12 = 1

2n

∫Sn−1

(R1� · R2� − n(� · e)(� · e))dθ .

To this aim, decompose Rn into the direct sum of E+ ⊕ E− ⊕ (

⊕i Ei ) where E±

are the eigenspaces of R−12 R1 corresponding to the eigenvalues ±1 and all E i are

two-dimensional vector spaces on which the restriction of R−12 R1 is a rotation of

angle θi ∈ (0, π). Since all these spaces are mutually orthogonal, it is easy tocheck that

γ12 = − ωn

n2n−1

(dim E− +

∑j

(1 − cos θj )

),

where ωn := |Sn−1|. Thus, the matrix � is invertible except when R1 = R2, andthis implies that (H2) holds whenever R1 �= R2. Observe that in this case we have


γ12 < 0. Finally, in order for (H3) to be fulfilled, it remains to check that it ispossible to choose A0 ∈ Mn(R) such that

λj := −∫

Sn−1

A0� · Rj� dθ < 0

for j = 1, 2. To this aim, we define

L : A0 ∈ Mn(R) −→ ∈ R2 .

This linear map is easily seen to be nonzero, so either it is surjective or its imageis included in a one-dimensional space. Assume that the latter is true. Then thereexists (α, β) �= (0, 0) such that

α

∫Sn−1

A0� · R1� dθ = β

∫Sn−1

A0� · R2� dθ

for all A0. Using this equality for A0 = R1 and A0 = R2, we conclude thatnecessarily α = ±β and ∫

Sn−1

R1� · R2� dθ = ±ωn .

However, a direct computation shows that∫Sn−1

R1� · R2� dθ = ωn

(1 − 2

ndim E− − 2

n

∑j

(1 − cos θj )

).

Since we already know that this quantity is in absolute value strictly less than ωn

when R1 �= R2, this implies that L is surjective and hence L−1(−∞, 0)2 is anopen nonempty set in Mn(R).

To summarize, we have obtained the following lemma:

LEMMA 2.1 Assume that e ∈ Rn − {0} and R1,R2 ∈ O(n) are chosen so that

R1e = R2e = e ;then (H1) holds. If in addition R1 �= R2, then (H2) holds, and the set of A0 suchthat (H3) holds is an open nonempty subset of Mn(R).

3 The Connection Laplacian on Tangent and Normal Bundles

We define some differential operators that appear naturally in the study of min-imal submanifolds. We also recall some well-known properties of these operators.This section is essentially borrowed from [13, 14], where further details and proofscan be found.


3.1 First-Order Differential OperatorsTo begin with, let us define the connections on the tangent bundle and on the

normal bundle of an m-dimensional submanifold M of some Euclidean space. LetV be any vector field on a submanifold M , and e be a tangent vector field on M .We will denote by ∇eV the full derivative of V along e.

The covariant derivative on the tangent bundle ∇ τ applied to the tangent vectorfield T and computed along the tangent vector e is defined to be the orthogonalprojection of ∇eT on the tangent bundle. That is,

∇τe T := [∇eT ]τM

where [ · ]τM denotes the orthogonal projection over the tangent bundle of M .Finally, we define the covariant derivative on the normal bundle ∇ ν , applied

to the normal vector field V and computed along the tangent vector e, to be theorthogonal projection of ∇e N on the normal bundle. That is,

∇νe N := [∇e N ]νM ,

where [ · ]νM denotes the orthogonal projection over the normal bundle of M .Let (e1, . . . , em) be a local orthonormal tangent frame field on M . For any

function f defined on M , we set

gradM f :=m∑

j=1

(ej f )ej ,

and for any tangent vector field T

divM T :=m∑

j=1

∇τej

T · ej .

3.2 Second-Order Differential OperatorsGiven (e1, . . . , em), a local orthonormal tangent frame field on M , we can de-

fine �M , the Laplace operator on M acting on functions, by

�M f :=m∑

j=1

(e2

j f − (∇τej

ej ) f).

The connection Laplacian on the tangent bundle of M , acting on tangent vectorfields, is defined by

�τM T :=

m∑j=1

(∇τej∇τ

ejT − ∇τ

∇τej

ejT

).

Let us recall that the operator �τM is a negative self-adjoint operator [14].


Finally, the connection Laplacian operator on the normal bundle of M , actingon normal vector fields, is defined by

�νM N :=

m∑j=1

(∇νej∇ν

ejN − ∇ν

∇τej

ejN

).

Let us recall that the operator �νM is a negative self-adjoint operator [8]. Naturally,

all these definitions do not depend on the choice of the local orthonormal tangentframe field (e1, . . . , em).

3.3 Zeroth-Order OperatorsThe second fundamental form of the m-dimensional submanifold M is a section

of the bundleT �(M) ⊗ T �(M) ⊗ N (M) ,

which is defined byBV,W := ∇ν

V W .

In other words, at any point p ∈ M , Bp represents a symmetric bilinear map fromTp(M) ⊗ Tp(M) into Np(M).

Given any local orthonormal tangent frame field (e1, . . . , em) on M , we definethe linear operator acting on normal vector fields

B(N ) :=m∑

j, j ′=1

(∇νej

ej ′ · N)∇ν

ejej ′ .

4 Differential Forms

It will be convenient to translate some of the previously defined operators in thelanguage of differential forms. Let p(M) denote the space of p-forms on M . Wedenote by d p the exterior derivative acting on p-forms

d : p(M) −→ p+1(M) ,

and define the operator

δ : p(M) −→ p−1(M) by δ := (−1)d(p+1)+1 � d� ,

where � denotes the Hodge operator.The Hodge Laplacian acting on p-forms is defined by

�p := −(dδ + δd) .

Observe that, with our sign conventions, �p is a negative operator.The inner product on p(M) is defined by

〈ω, ω〉�p(M) :=∫M

ω ∧ �ω .


One can check that, given the definition of δ, for any (p − 1)-form ω and anyp-form ω, we have ∫

M

dω ∧ �ω =∫M

ω ∧ �δω ,

and hence, for any p-forms ω and ω, we have∫M

�pω ∧ �ω = −∫M

(dω ∧ �dω + δω ∧ �δω) .

One can identify functions on M with 0-forms in the obvious way and, by using themetric on M , there is a natural identification of tangent vector fields with 1-forms.

Indeed, let us assume that we have chosen a local orthonormal tangent framefield (e1, . . . , em) at a point p ∈ M and that the metric is given by g = (gi j )i, j with

ej = 1√gj j

∂

∂x j

at p. Then, to any tangent vector

T =m∑

j=1

Ti ei ,

we can associate the 1-form

ω =m∑

j=1

√gj j Tj dx j .

Observe that this identification is consistent with the definitions of the inner prod-ucts on 1(M) and on T (M); namely, if the vector field T is associated to the1-form ω, we have

‖T ‖2L2(M)

:=∫M

|T |2 =∫M

ω ∧ �ω := ‖ω‖2 .

Given this identification of vector fields with 1-forms, we can identify �1, theHodge Laplacian on 1-forms, with some symmetric second-order differential oper-ator acting on tangent vector fields. We will still denote by �1 this operator, whichis now defined on tangent vector fields. The relation between �1 and the connec-tion Laplacian on the tangent bundle that we have defined in the previous sectionis given by the following:

THEOREM 4.1 [14] The difference between the Hodge Laplacian on 1-forms andthe connection Laplace operator on the tangent bundle is given by

�1 = �τM − Ric ,

where Ric denotes the Ricci tensor.


5 The Hyperbola

The n-dimensional hyperbola HI is parametrized by

X :=(

cos s(sin(ns))1/n

�,sin s

(sin(ns))1/n�

),

where s ∈ (0, πn ) and where � is a parametrization of the (n − 1)-dimensional

sphere Sn−1.

It will also be convenient to identify R2n with C

n using

Rn × R

n � (x, y) ∼ x + iy ∈ Cn .

With this identification in mind, we will write

X := eis

(sin(ns))1/n� .

5.1 The Tangent BundleThe tangent space of HI is spanned by the following set of vectors:

∂s X = − ei(1−n)s

(sin(ns))1+1/n�,

and, for all j = 1, . . . , n − 1,

∂θj X = eis

(sin(ns))1/n∂θj � .

In order to simplify the notation, we will write

�j := ∂θj �,

and we will define the vectors

e0 := ei(1−n)s� and ej := eis �j

|�j | .

It will be convenient to assume that, at a given point on the sphere, the parametriza-tion � is chosen in such a way that

(5.1) ∂θj � · ∂θk � = 0 if j �= k .

In this case, (e0, . . . , en−1) is an orthonormal basis of T (HI ) at the point X .Finally, we define, for all j = 1, . . . , n − 1,

εj := �j

|�j | ,

so that, when (5.1) is satisfied, (ε1, . . . , εn−1) is an orthonormal basis of T (Sn−1)

at the point �.


5.2 The Normal BundleThe normal space to HI is spanned by the following vectors:

N0 := iei(1−n)s�

and, for j = 1, . . . , n − 1,

Nj := ieis �j

|�j | .

Hence any normal vector field on HI can be written as

N := iei(1−n)s f � + ieis T ,

where f is a real-valued function and, for all s ∈ (0, πn ), T (s, ·) is a tangent vector

field on Sn−1.

6 Expansion of the Lower End of the Hyperbola

We define a new variable r > 0 by

r := cos s(sin(ns))1/n

.

As s tends to 0, the expansion of r in terms of s is given by

r = (ns)−1/n(1 + O(s2)) .

In turn, we get the expansion

sin s(sin(ns))1/n

= r1−n

n(1 + O(r−2n))

as r tends to +∞. The hyperbola defined in the previous section can be parame-trized using the parameter r instead of s. As s tends to 0, or as r tends to +∞,we obtain the expansion of the parametrization of the lower end of the hyperbola,namely,

X =(

r + ir1−n

n(1 + O(r−2n))

)�

as r tends to +∞.We now consider the hyperbola scaled by a factor (nβε)1/n for some β > 0 and

ε > 0, namely,

Xβ = (nβε)1/n eis

(sin(ns))1/n� .

We define the variable r := (nβε)1/nr and, as s tends to 0, or r tends to +∞, weobtain the expansion

(6.1) Xβ = (r + iεβr1−n(1 + O(ε2r−2n))

)� .


7 The Jacobi Operator about the Hyperbola

We recall from [8] that the Jacobi operator, which is also called the linearizedmean curvature operator, is given by

L H := �νH + B .

In order to compute the Jacobi operator about the hyperbola, we use the definitionsand results of Section 3. We find that, in the local chart given by X , the followingholds:

PROPOSITION 7.1 The Jacobi operator about the hyperbola HI reads

(sin(ns))−2/n L H N =iei(1−n)s[(sin(ns))2−2/n∂s

((sin(ns))2/n∂s f

) + �S f − 2 cos(ns) divS T

+ (n2 − 1) sin2(ns) f − (n − 1) f]�

+ ieis[(sin(ns))2−2/n∂s((sin(ns))2/n∂s T

) + �τST + 2 cos(ns) gradS f

− T + 3 sin2(ns)T],

where N = iei(1−n)s f � + ieis T ∈ N (HI ).

In order to study the spectral properties of L H , we identify tangent vector fieldson S

n−1 with 1-forms. This identification yields a natural identification of the nor-mal vector field on HI

N = iei(1−n)s f � + ieis T with U := ( f, v) ,

where v(s, ·) is the 1-form corresponding to T (s, ·).Given this identification, we use the results of Section 4 and identify the Jacobi

operator L H with the following linear second-order operator:

(sin(ns))−2/n L H ( f, v) = (sin(ns))2−2/n∂s((sin(ns))2/n∂s( f, v)

) + (�0 f,�1v)

+ 2 cos(ns)(δv, d f ) + ((1 − n) f, (n − 3)v)

+ sin2(ns)((n2 − 1) f, 3v) ,

where the operators d, δ, �0, and �1 are defined as in Section 4, where it is under-stood that the background manifold is M = S

n−1.In order to have a better understanding of the structure of this operator, we

define the variable t by

e−nt = sin(ns)1 − cos(ns)

.

Observe that

dt = 1

sin(ns)ds ,

and we also have

sin(ns) = (cosh(nt))−1 and cos(ns) = − tanh(nt) .


We define the conjugate operator

LH := (sin(ns))−(n+2)/2n L H (sin(ns))(n−2)/2n .

Using the previous identities, one checks that we have

LH ( f, v) = ∂2t ( f, v) + (�0 f,�1v) − 2 tanh(nt)(δv, d f )

− 1

4(n2 f, (n − 4)2v) + 1

4 cosh2(nt)(3n2 f, (16 − n2)v) .

Because the operators L H and LH are conjugate, their mapping properties are iden-tical. Observe that, as the parameter t tends to −∞, the operator LH is asymptoticto the following differential operator:

L0( f, v) := ∂2t ( f, v) + (�0 f,�1v) + 2(δv, d f ) − 1

4(n2 f, (n − 4)2v) .

The forthcoming sections are devoted to the study of the mapping properties ofboth LH and L0.

8 Eigendata of �0 and �1 on Sn−1

We recall some well-known facts about the eigendata of �0 and �1 when thebackground manifold is S

n−1. Further information and proofs can be obtainedin [5, 6, 17].

To begin with, recall that the spectrum of �0 on Sn−1 is given by

σ(�0) = {k(n − 2 + k) : k ≥ 0} .

Now, in dimension n = 2 or n = 3 the spectrum of �1 on Sn−1 is simply given by

σ(�1) = {k(n − 2 + k) : k ≥ 1} ,

and all 1-eigenforms of �1 are images of eigenfunctions of �0 by the operator d.In dimension n ≥ 4, things are slightly more involved. The spectrum of �1 can

be decomposed into two disjoint subsets

σ(�1) = σex(�1) ∪ σcoex(�

1) ,

whereσex(�

1) = {k(n − 2 + k) : k ≥ 1}corresponds to the eigenvalues associated to exact 1-eigenforms (i.e, 1-eigenformsthat belong to the image of 0(Sn−1) by d) and where

σcoex(�1) = {(k + 1)(n − 3 + k) : k ≥ 1}

corresponds to the eigenvalues associated to coexact 1-eigenforms (i.e., 1-eigen-forms that belong to the image of 2(Sn−1) by δ). It is known that all exact1-eigenforms are the images of eigenfunctions of �0 by d, and all coexact 1-eigenforms are the images of exact 2-eigenforms of �2 by δ.

We defineV0 := {( f, 0) : d f = 0} .


Next, for all k ≥ 1, we define

Vkex := {( f, v) : (�0 f,�1v) = −k(n − 2 + k)( f, v) and dv = 0} ,

the eigenspace corresponding to the eigenvalue k(n − 2 + k) and to exact 1-forms,and

Vkcoex := {(0, v) : �1v = −(k + 1)(n − 3 + k)v and δv = 0} ,

the eigenspace corresponding to the eigenvalue k(n−2+k) and to coexact 1-forms.This decomposition corresponds to the decomposition of 1-forms on S

n−1 givenby Hodge’s decomposition theorem [2]

1(Sn−1) = d 0(Sn−1) ⊕ δ 2(Sn−1) ,

since there are no harmonic 1-forms on Sn−1. In addition, the two spaces d 0(Sn−1)

and δ 2(Sn−1) are orthogonal. This decomposition of the space of 1-forms inducesa natural decomposition of the space of tangent vector fields into (1) vector fieldscorresponding to elements of d 0(Sn−1) and (2) vector fields corresponding to el-ements of δ 2(Sn−1). These two sets of vector fields are orthogonal in the L 2

sense.

9 Indicial Roots

In this section, we compute the indicial roots corresponding to the operator LH .These indicial roots determine the asymptotic behavior, as t tends to ±∞, of thesolutions of the homogeneous problem LHU = 0.

Observe that, in dimension n ≥ 4, we can decompose any 1-form v = vex +vcoex into the sum of an exact 1-form on S

n−1 and a coexact 1-form on Sn−1. In

dimension n = 2 and n = 3, we set vcoex = 0. Now, if U = ( f, v) is a solution ofthe homogeneous equation LH U = 0, then vcoex satisfies LH (0, vcoex) = 0; hence

(9.1) ∂2t vcoex + �1vcoex − (n − 4)2

4vcoex + 16 − n2

4

vcoex

cosh2(nt)= 0 .

However, we have seen in Section 8 that, when it is restricted to coexact forms, thespectrum of �1 is given by the set of (k + 1)(n − 3 + k) for k ≥ 1. In order to findthe indicial roots corresponding to this operator, we look for solutions of (9.1) ofthe form

vcoex = aψ ,

where ψ is a coexact eigenform of �1 corresponding to the eigenvalue (k +1)(n −3 + k) and where a is a scalar function depending only on t . Therefore, the scalarfunction a is a solution of the following ordinary differential equation:

(9.2) a −(

n − 2

2+ k

)2

a + 16 − n2

4

a

cosh2(nt)= 0 .


The behavior of the solutions of (9.2), as t tends to either ±∞, is given by eγ ±k t

where

γ ±k = ±

(n − 2

2+ k

).

The constants γ ±k , for k ≥ 1, are the indicial roots of the operator LH when it is

restricted to coexact 1-forms.We now turn to the study of indicial roots of the operator LH , when it is re-

stricted to exact 1-forms. Observe that this time the equation LH ( f, vex) = 0yields the coupled system

(9.3) ∂2t ( f, vex) + (�0 f,�1vex) − 2 tanh(nt)(δvex, d f )

− 1

4(n2 f, (n − 4)2vex) + 1

4 cosh2(nt)(3n2 f, (16 − n2)vex) = 0 .

Now, according to the analysis of Section 8, if φ is an eigenvalue of �0 associ-ated to the eigenvalue k(n − 2 + k), we look for solutions of (9.3) of the form

f = aφ and vex = bdφ ,

where a and b are scalar functions depending only on t . This yields the followingordinary differential equation:

a − n2

4a + 3n2

4

acosh2(nt)

= 0

when k = 0, and the system of ordinary differential equations

a − k(n − 2 + k)a − n2

4a − 2k(n − 2 + k) tanh(nt)b + 3n2

4

acosh2(nt)

= 0

b − k(n − 2 + k)b − (n − 4)2

4b − 2 tanh(nt)a + 16 − n2

4

b

cosh2(nt)= 0

when k �= 0. With little work, we find that, when k = 0, the asymptotic behaviorof a at both ±∞ is governed by the exponents

µ±0 = ±n

2,

and, when k �= 0, the asymptotic behavior of a and b at both ±∞ is governed bythe following sets of indicial roots:

µ±k = ±

(n2

+ k)

and ν±k = ±

(n − 4

2+ k

).

At this point, it is worth mentioning that the operators LH and L0 have the sameindicial roots.


10 The Jacobi Fields

Jacobi fields are solutions of the homogeneous problem L H U = 0. While theexplicit expression of the Jacobi fields is in general impossible to obtain, some Ja-cobi fields can be explicitly computed since they correspond to geometric transfor-mations of the hyperbola. Observe that, in this paragraph, we consider the Jacobioperator L H (instead of the conjugate operator LH ), and we express everything interms of the variable s (instead of t), since the geometric Jacobi fields are easier todescribe for the operator L H using the variable s.

In the remainder of this paper, only the Jacobi fields corresponding to transla-tions and dilations will be useful. However, for the sake of completeness, we havecollected all the geometric Jacobi fields as well as their explicit expression.

Jacobi Fields Corresponding to Translations. Let a ∈ Rn and α ∈ R be given;

the Jacobi field corresponding to the translation of vector eiαa is the projectionover the normal bundle of the constant Killing vector field K (x + iy) = eiαa. Weobtain explicitly

(10.1) �t(α, a) = sin((n−1)s+α)(a·�)iei(1−n)s�+sin(α−s)ieis(a−(a·�)�) .

Jacobi Field Corresponding to Dilations. The Jacobi field corresponding to di-lations is the projection over the normal bundle of the Killing vector field K (x +iy) = x + iy. We obtain

(10.2) �d = (sin(ns))1−1/niei(1−n)s� .

Jacobi Fields Corresponding to the Action of SU(n). Denote by A an n × nsymmetric matrix. The Jacobi fields corresponding to the action of

x + iy ∈ Cn −→ ei A(x + iy) ∈ C

n

is the projection of the Killing vector field K (x + iy) = i A(x + iy) over the normalbundle. We get explicitly

�SU(n)(A) =(sin(ns))−1/n( cos(ns)(A� · �)iei(1−n)s� + ieis(A� − (A� · �)�)

).

Jacobi Fields Corresponding to the Action of O(2n)/ SU(n). Let A be somen × n skew-symmetric matrix. The Jacobi fields corresponding to the action of

x + iy ∈ Cn −→ (e−Ax + ieA y) ∈ C

n

is the projection of the Killing vector field K (x + iy) = A(x − iy) over the normalbundle. We obtain explicitly

(10.3) �O(2n)(A) = (sin(ns))−1/n sin(2s)ieis A� .

The Jacobi field corresponding to the action of

x + iy ∈ Cn −→ (cosh A(x + iy) + sinh A(y + i x)) ∈ C

n


is the projection of the Killing vector field K (x + iy) = i A(x − iy) over the normalbundle. We explicitly obtain

�O(2n)(A) = (sin(ns))−1/n cos(2s)ieis A� .

11 Injectivity Results

In this section, we prove various injectivity results for the Jacobi operator LH

and also for the operator L0, which appears as the limit of LH when t tends to −∞.Basically, we would like to prove that these operators are injective (when definedover some appropriate space of functions). This will be rather easy to prove forthe operator L0. However, the proof of the injectivity result turns out to be ratherdelicate for the operator LH .

To begin with, we prove the following:

PROPOSITION 11.1 Assume that U is a solution of L0U = 0 in (t1, t2)×Sn−1 (with

U = 0 on the boundary if t1 > −∞ or t2 < +∞). Furthermore, assume that

|U | ≤ (cosh t)δ

for some δ < (n − 2)/2. Then U ≡ 0.

PROOF: The proof of this result can be obtained first by decomposing U overthe spaces V0, V0

coex, and Vkex, which correspond to the eigenspaces of (�0,�1),

and then by computing explicitly the solutions of the ordinary differential equationassociated to each projection, and finally by showing that the growth restrictionimplies that all the components of the eigenfunction decomposition of U are equalto 0.

However, since this argument will not generalize easily to the study of theoperator LH , we will proceed somewhat differently. We decompose U over theeigenspaces of (�0,�1), namely,

U = U0 +∑k≥1

(U k

ex + U kcoex

),

where U0 ∈ V0, U kex ∈ Vk

ex, and U kcoex ∈ Vk

coex.In the case where t1 = −∞ or t2 = +∞, we will need to justify all the forth-

coming integrations by parts. But the inspection of the indicial roots of L0, togetherwith the fact that the solution U is assumed to be bounded by (cosh t)δ for someδ < (n − 2)/2, implies that the projection of U over V 0, Vk

ex, or Vkcoex is actually

exponentially decaying at infinity. Hence, the forthcoming analysis can be appliedto any of the components of the eigenfunction decomposition of U , and this isenough to show that U ≡ 0.

Step 1. We multiply the equation L0U = 0 by U kcoex := (0, vcoex) and integrate

by parts the result over (t1, t2) × Sn−1. We obtain∫

|∂tvcoex|2 +∫

|dvcoex|2 + (n − 4)2

4

∫|vcoex|2 = 0 .


Furthermore, the first eigenvalue of �1 acting on coexact 1-forms being equal to2(n − 2), we have ∫

Sn−1

|dvcoex|2 ≥ 2(n − 2)

∫Sn−1

|vcoex|2 .

Integrating this inequality over t , we get∫|dvcoex|2 ≥ 2(n − 2)

∫|vcoex|2 .

Hence, we conclude that∫|∂tvcoex|2 + n2

4

∫|vcoex|2 ≤ 0 ,

which immediately implies that vcoex ≡ 0. Hence U kcoex = 0 for all k ≥ 1.

Step 2. We multiply the equation L0U = 0 by U kex := ( f, vcoex) and integrate

by parts over (t1, t2) × Sn−1. This time, we obtain

(11.1)∫ (|∂t f |2 + |∂tvex|2

) +∫ (|d f |2 + |δvex|2

)+ 1

4

∫ (n2 f 2 + (n − 4)2|vex|2

) = 2∫

(d f ∧ �vex + f δvex) .

Using the Cauchy-Schwarz inequality, we estimate∣∣∣∣∫

d f ∧ �vex

∣∣∣∣ ≤(∫

|d f |2)1/2( ∫

|vex|2)1/2

,

∣∣∣∣∫

f δvex

∣∣∣∣ ≤(∫

|δvex|2)1/2( ∫

f 2)1/2

.

(11.2)

To estimate the right-hand side of the first inequality in (11.2), we use 2ab ≤a2 + b2, and, in order to estimate the right-hand side of the second inequality in(11.2), we use 2ab ≤ 1

2 a2 + 2b2.Collecting the previous inequalities and considering (11.1), we conclude that

(11.3) C + n2

4A + C ′ + (n − 4)2

4A′ ≤ 1

2C ′ + 2A + C + A′ ,

where, in order to simplify the notation, we have set

A :=∫

f 2 , B :=∫

|∂t f |2 , C :=∫

|d f |2,

A′ :=∫

|vex|2 , B ′ :=∫

|∂tvex|2 , C ′ :=∫

|δvex|2 .


To complete the proof, we use the fact that the first eigenvalue of �1 acting onexact 1-forms is equal to (n − 1); hence we have∫

Sn−1

|δvex|2 ≥ (n − 1)

∫Sn−1

|vex|2 .

Integration of this inequality over t implies that∫|δvex|2 ≥ (n − 1)

∫|vex|2 ,

otherwise stated C ′ ≥ (n − 1)A′. This, together with (11.3), implies that

n2 − 8

4A + (n − 4)2 + 2n − 6

4A′ ≤ 0 ,

which proves that f ≡ 0 and vex ≡ 0. Hence U kex = 0 for all k ≥ 1.

Step 3. Finally, we multiply the equation L0U = 0 by U 0 := ( f, 0) and inte-grate by parts over (t1, t2) × S

n−1. We obtain∫|∂t f |2 + n2

4

∫f 2 = 0 .

This implies that f ≡ 0. Hence U 0 = 0. The proof of the result is thereforecomplete. �

We would like to prove a similar result for the Jacobi operator. However, theinjectivity result is not true in such generality. Indeed, the Jacobi field �d cor-responding to dilations, which has been described in Section 10, gives rise toU := ( f0, 0) with

f0(t) := (cosh(nt))−1/2 ,

which is a solution of LHU = 0 in R × Sn−1. Observe that this solution decays

exponentially at both ±∞. Nevertheless, we have the following proposition:

PROPOSITION 11.2 Assume that n ≥ 4 and assume that U is a solution of LH U =0 in (t1, t2)×S

n−1 (with U = 0 on the boundary if t1 > −∞ or t2 < +∞). Furtherassume that

|U | ≤ (cosh t)δ

for some δ < (n − 2)/2 and that, for every t ∈ (t1, t2), U (t, ·) is orthogonal to V0

in the L2 sense on Sn−1. Then U ≡ 0.

PROOF: Again, we decompose U over the eigenspaces of (�0,�1), namely,

U =∑k≥1

(U k

ex + U kcoex

),

where U kex ∈ Vk

ex and U kcoex ∈ Vk

coex. Observe that, by assumption, there is nocomponent over V0. The indicial roots of L0 and LH being identical, we can argueas in the previous proof to justify the integrations by parts when t1 = −∞ ort2 = +∞.


Step 1. We multiply the equation LHU = 0 by U kcoex := (0, vcoex) and integrate

over (t1, t2) × Sn−1 to obtain∫

|∂tvcoex|2 +∫

|dvcoex|2 + (n − 4)2

4

∫|vcoex|2 =

16 − n2

4

∫(cosh(nt))−2|vcoex|2 .

Furthermore, as in step 1 in the proof of Proposition 11.1, we have∫|dvcoex|2 ≥ 2(n − 2)

∫|vcoex|2 .

Hence ∫|∂tvcoex|2 + n2

4

∫|vcoex|2 ≤ 16 − n2

4

∫(cosh(nt))−2|vcoex|2 ,

which immediately implies that vcoex ≡ 0. Hence U kcoex = 0 for all k ≥ 1.

Step 2. We multiply the equation LH U = 0 by U kex := ( f, vex) and integrate

by parts over (t1, t2) × Sn−1. This time, we obtain∫ (|∂t f |2 + |∂tvex|2

) +∫ (|d f |2 + |δvex|2

) +∫ (

n2

4f 2 + (n − 4)2

4|vex|2

)=

∫(cosh(nt))−2

(3n2

4f 2+16 − n2

4|vex|2

)−2

∫tanh(nt)(d f ∧�vex+ f δvex) .

Because the proof is quite involved and to simplify the notation, we set

A :=∫

f 2 , B :=∫

(cosh(nt))−2 f 2 , C :=∫

|∂t f |2,

A′ :=∫

|vex|2 , B ′ :=∫

(cosh(nt))−2|vex|2 , C ′ :=∫

|∂tvex|2 ,

D :=∫

|d f |2 , D′ :=∫

|δvex|2 ,

and

E :=∫

tanh(nt)d f ∧ �vex =∫

tanh(nt) f δvex .

With our notation, the equality we have obtained reads

(11.4) C + C ′ + D + D′ + n2

4A + (n − 4)2

4A′ − 3n2

4B + n2 − 16

4B ′ = −4E .

Using the Cauchy-Schwarz inequality and integration by parts, we estimate∣∣∣∣∫

tanh(nt) f δvex

∣∣∣∣ ≤(∫

|δvex|2)1/2(∫

f 2 −∫

(cosh(nt))−2 f 2)1/2

.

This inequality can also be written as

(11.5) E2 ≤ D′(A − B) .


As in step 2 of the proof of Proposition 11.1, we have∫|d f |2 ≥ (n − 1)

∫f 2 ,

otherwise stated

(11.6) D ≥ (n − 1)A .

Finally, we use an integration by parts to prove that

n∫

(cosh(nt))−2 f 2 = −2∫

tanh(nt) f ∂t f .

Using the Cauchy-Schwarz inequality, we conclude that

n∫

(cosh(nt))−2 f 2 ≤ 2

(∫|∂t f |2

)1/2(∫f 2 −

∫(cosh(nt))−2 f 2

)1/2

.

A similar inequality holds with f replaced by vex. This yields

(11.7) n2 B2 ≤ 4C(A − B) , n2 B ′2 ≤ 4C ′(A′ − B ′) .

Step 2.1. Assuming that A �= B and A′ �= B ′, we can collect the previousinequalities (11.5) through (11.7) to eliminate C , C ′, D, and D′ in (11.4). Withlittle work we find

n2

4

(2B − A)2

A − B+ 1

4

(4B ′ + (n − 4)A′)2

A′ − B ′ + (n − 5)A

+ 4B + (E + 2(A − B))2

A − B≤ 0 .

It is now an easy exercise to check that, when n ≥ 4, the above inequality impliesthat A = A′ = 0. Hence f ≡ 0 and vex ≡ 0.

Step 2.2. If A = B, we first use (11.7) to conclude that B = 0; hence weconclude that f ≡ 0. In this case, it readily follows from (11.4) that A′ = 0, whichimplies that vex ≡ 0.

Step 2.3. If A′ = B ′, we first use (11.7) to conclude that B ′ = 0, so vex ≡ 0. Inthis case, (11.4) reduces to

C + D + n2

4A − 3n2

4B = 0 .

Using (11.6) through (11.7), we conclude that A = 0, which implies that f ≡ 0.

In each case, we have shown that f ≡ 0 and vex ≡ 0; hence U kex = 0 for all

k ≥ 1. The proof of the result is therefore complete. �

When n = 3 some modifications are needed in the corresponding statement.Indeed, we prove—and this will be sufficient for our purposes—that the injectivityresult as stated in Proposition 11.2 holds on any interval containing a large intervalcentered at the origin. This additional assumption is probably not needed, but we


have not been able to get rid of it. Nevertheless, the result stated below is just whatwe need for all the remaining analysis to hold.

PROPOSITION 11.3 Assume that n = 3. Then there exists t0 > 0 such that, if U isa solution of LH U = 0 in (t1, t2) × S

2, for some t1 ≤ −t0 and t2 ≥ t0 (with U = 0on the boundary if either t1 > −∞ or t2 < +∞), which satisfies

|U | ≤ (cosh t)δ

for some δ < (n − 2)/2 = 12 , and with the property that, for every t ∈ (t1, t2),

U (t, ·) is orthogonal to V0 in the L2 sense on S2, then U ≡ 0.

PROOF: As in the previous proof, we decompose

U =∑k≥1

(U k

ex + U kcoex

).

Step 1. We proceed as in step 1 in the proof of the previous result to show thatU k

coex = 0 for all k ≥ 1.

Step 2. Let us show that, for all k ≥ 2, U kex = 0. Indeed, if k ≥ 2, then (11.6)

can be improved to

D ≥ 6A,

where 6 corresponds to the next eigenvalue (6 = 2n when n = 3), whereas wewould only have D ≥ 2A if we consider that k ≥ 1. Now, we can proceed as instep 2 of the proof of the previous result to show that U k

ex = 0. Observe that, forthe time being, no additional assumptions are needed.

Step 3. It just remains to prove that U 1ex = 0. In this case, since V1

ex is finitedimensional, we are reduced to studying some coupled system of ordinary differ-ential equations. Indeed, by decomposing U 1

ex(t, ·) over the eigenfunctions �0 and�1 that are associated to the eigenvalue 2, it is enough to consider the case where

U 1ex = (aφ, bdφ) ,

where a and b are scalar functions that depend only on t , and φ is an eigenfunc-tion of �0 corresponding to the eigenvalue 2. Since LHU 1

ex = 0, we obtain thefollowing system of ordinary differential equations for a and b:

(11.8)

a − 17

4a − 4 tanh(3t)b + 27

4

a

cosh2(3t)= 0

b − 9

4b − 2 tanh(3t)a + 7

4

b

cosh2(3t)= 0 .

As already mentioned, the asymptotic behavior of a and b at both ±∞ is gov-erned by the indicial roots of LH restricted to V1

ex. Namely,

µ±1 = ±5

2and ν±

1 = ±1

2.


Observe that we know explicitly one solution of (11.8), namely, the solutions cor-responding to the Jacobi field �t(α, a); see Section 10. In particular, when α = 0,we obtain the solution (a1, b1) of (11.8),

(a1, b1) = (sin(3s))−1/6(sin(2s),− sin s) .

Recall that s is a function of t and that we have sin(3s) = (cosh(3t))−1 andcos(3s) = − tanh(3t). Using this, we can easily obtain the asymptotic behavior ofthis explicit solution. Near −∞ it is given by

(a1, b1) ∼ (2(cosh t)−5/2,−(cosh t)−5/2) ,

and, near +∞, it is given by

(a1, b1) ∼ ((cosh t)1/2,−(cosh t)1/2) ,

where A ∼ B means that A = cB(1+o(1)) at ∞ for some positive constant c > 0.Now observe that there exists another independent solution of (11.8) that behaveslike

(a2, b2) ∼ ((cosh t)−1/2,−(cosh t)−1/2)

near +∞. In order to determine the asymptotic behavior of this solution as t tendsto −∞, we make use of the fact that

(a1a2 − a2a1) + 2(b1b2 − b2b1)

is a constant function, as can be checked by differentiation with respect to t . Eval-uating this quantity at +∞ and at −∞, one proves that

(a2, b2) ∼ (−2(cosh t)5/2, (cosh t)5/2)

near −∞.Finally, observe that, if (a, b) is a solution of (11.8), then

t −→ (a(−t),−b(−t))

is also a solution of (11.8). Applying this recipe to both (a1, b1) and (a2, b2) yieldsthe last two independent solutions of (11.8).

Since any solution of (11.8) is a linear combination of these four indepen-dent solutions, it is now a simple exercise to show that there are no solutions of(11.8) that are defined in R and bounded by (cosh t)δ for some δ < 1

2 . With littlemore work, one also shows that there are no solutions of (11.8) on (t1, t2) that arebounded by (cosh t)δ, provided −t1 and t2 are chosen large enough. This completesthe proof of the result. �

12 Mapping Properties of LH

In this section we perform the analysis of the mapping properties of the Jacobioperator LH . To begin, we define the weighted Hölder spaces we will work with.


DEFINITION 12.1 For all δ ∈ R and for all t0 ∈ R, the space Ck,αδ ([t0,+∞) ×

Sn−1; 0 × 1) is defined to be the space of U ∈ Ck,α([t0,+∞) × S

n−1; 0 × 1)

for which the following norm is finite:

‖U‖k,α,δ := supt≥t0

‖e−δsU‖k,α,([t,t+1]×Sn−1) .

Here ‖ · ‖k,α,([t,t+1]×Sn−1) denotes the usual Hölder norm on [t, t + 1] × Sn−1.

Observe that U = ( f, v) where v is a 1-form; hence, in the above-defined normit is the coefficients of v and the function f that are estimated.

The mapping properties of LH when defined between the weighted spaces de-scribed above crucially depend on the choice of the weight parameter δ. We provethe following proposition:

PROPOSITION 12.2 Assume that δ ∈ ((2 − n)/2, (n − 2)/2) and α ∈ (0, 1) arefixed and that there exists some constant c > 0. In addition, for all t0 ∈ R (whenn = 3, we further assume that t0 < −t0, where t0 has been defined in Proposi-tion 11.3), there exists an operator

Gt0 : C0,αδ

([t0,+∞) × Sn−1; 0 × 1) −→ C2,α

δ

([t0,+∞) × Sn−1; 0 × 1)

such that, for all V ∈ C0,αδ ([t0,+∞) × S

n−1; 0 × 1), if, for all t > t0, V (t, ·) isorthogonal to V0 in the L2 sense on S

n−1, then U = Gt0 V is the unique solution of{LH U = V in [t0,+∞) × S

n−1

V = 0 on {t0} × Sn−1 ,

which belongs to C2,αδ ([t0,+∞) × S

n−1; 0 × 1). Furthermore, ‖U‖2,α,δ ≤c‖V ‖0,α,δ.

PROOF: Uniqueness of Gt0 follows Proposition 11.2 (or Proposition 11.3 whenn = 3). We therefore concentrate our attention on the existence of Gt0 and thederivation of the uniform estimate for its norm.

Since minor modifications are needed to treat the case n = 3, we shall assumefrom now on that n ≥ 4 in order to simplify the discussion. It follows from Propo-sition 11.2 that, when restricted to the space of U , which is orthogonal to V 0 inthe L2 sense on S

n−1, the operator LH is injective over (t ′, t ′′) × Sn−1. As a con-

sequence, for all t ′′ > t ′, we are able to solve LHU = V in (t ′, t ′′) × Sn−1, with

U = 0 on {t ′, t ′′} × Sn−1.

We claim that, there exists some constant c > 0 independent of t ′′ > t ′ + 1 andof V such that

sup(t ′,t ′′)×Sn−1

|e−δtU | ≤ c sup(t ′,t ′′)×Sn−1

|e−δt V | .We argue by contradiction and assume that the claim is not true. In this case, therewould exist sequences t ′′

i > t ′i + 1, a sequence Vi satisfying

sup(t ′i ,t ′′i )×Sn−1

|e−δt Vi | = 1 ,


and a sequence Ui of solutions of LHUi = Vi in (t ′i , t ′′

i ) × Sn−1 with Ui = 0 on

{t ′i , t ′′

i } × Sn−1 such that

Ai := sup(t ′i ,t ′′i )×Sn−1

|e−δtUi | −→ +∞ .

Furthermore, Ui (t, ·) and Vi (t, ·) are orthogonal to V0 in the L2 sense on Sn−1.

Let us denote by (ti , θi ) ∈ (t ′i , t ′′

i ) × Sn−1 a point where the above supremum

Ai is achieved. We now distinguish a few cases according to the behavior of thesequence ti (which, up to a subsequence, can always be assumed to converge in[−∞,+∞]). Up to some subsequence, we may also assume that the sequencet ′′i − ti (respectively, t ′

i − ti ) converges to t∗ ∈ (0,+∞] (respectively, to t∗ ∈[−∞, 0)).

At this point, observe that the sequence t ′i − ti remains bounded away from 0.

Indeed, since Ui and LHUi are bounded by a constant (independent of i) timeseδt ′i Ai in [t ′

i , t ′i + 1] × S

n−1, and since Ui = 0 on {t ′i } × S

n−1, standard ellipticestimates allow us to conclude that ∂tUi is also uniformly bounded by a constanttimes eδt ′i Ai in [t ′

i , t ′i + 1

2 ] × Sn−1. As a consequence, the above supremum cannot

be achieved at a point that is too close to t ′i . Similarly, one proves that the sequence

t ′′i − ti also remains bounded away from 0.

We now define the rescaled sequence

Ui (t, θ) := e−δti

AiUi (t + ti , θ) .

Case 1. Assume that the sequence ti converges to t∞ ∈ R. After the extractionof some subsequences (if this is necessary), we may assume that the sequenceeδti Ui (· − ti , ·) converges to some nontrivial solution of

(12.1) LH U∞ = 0

in (t∗ + t∞, t∗ + t∞) × Sn−1, with boundary condition U∞ = 0, if either t∗ or t∗ is

finite. Furthermore, by construction we have

(12.2) sup(t∗+t∞,t∗+t∞)×Sn−1

|e−δtU∞| = 1 ,

and U∞(t, ·) is orthogonal to V0 in the L2 sense on Sn−1. We define U k

ex (respec-tively, U k

coex) to be the projection of U∞ over the space Vkex (respectively, Vk

coex).This yields the decomposition

U∞ =∑k≥1

(U k

ex + U kcoex

).

If t∗ = −∞, an inspection of the indicial roots of LH shows that U kex and U k

coex arebounded by a constant times e

n−22 t at −∞, since they are known to be bounded by

eδt for some δ ∈ ( 2−n2 , n−2

2 ). Similarly, if t∗ = +∞, U kex and U k

coex are bounded by a

constant times e2−n

2 t at +∞, since they are bounded by eδt for some δ ∈ ( 2−n2 , n−2

2 ).


Applying the result of Proposition 11.2, we conclude that U kex = 0 and U k

coex = 0;hence U∞ = 0, which contradicts (12.2).

Case 2. Assume that the sequence ti converges to −∞. After the extractionof some subsequences (if this is necessary), we may assume that the sequence Ui

converges to some nontrivial solution of

(12.3) L0U∞ = 0

in (t∗, t∗) × Sn−1, with boundary condition U∞ = 0, if either t∗ or t∗ is finite.

Furthermore,

(12.4) sup(t∗,t∗)×Sn−1

|e−δtU∞| = 1 .

As already mentioned, the indicial roots of the operator L0 are the same as theindicial roots corresponding to LH . Again, the inspection of the indicial roots ofthe operator L0 shows that the projection of U∞ over the space Vk

ex or over thespace Vk

coex decays exponentially at ∞, if either t∗ = −∞ or t∗ = +∞. This time,we apply the result of Proposition 11.1 to conclude that U∞ = 0, contradicting(12.4).

Case 3. Assume that the sequence ti converges to +∞. Because this case issimilar to case 2, we omit it.

Since we have ruled out every possibility, the proof of the claim is complete.Now we can pass to the limit t ′′ → +∞ in a sequence of solutions defined on(t ′, t ′′) × S

n−1 and obtain a solution of LH U = V , which is defined in (t ′,+∞) ×S

n−1, with U = 0 on {t ′} × Sn−1, and which satisfies

sup(t ′,+∞)×Sn−1

|e−δtU | ≤ c sup(t ′,+∞)×Sn−1

|e−δt V |

for some constant c > 0 independent of t ′. To complete the proof of the proposi-tion, it suffices to apply Schauder’s estimates [7] in order to get the relevant esti-mates for all the derivatives of U . We leave the details to the reader. �

We now extend the right inverse Gt0 to the set of V := (g, 0) where g dependsonly on t . For simplicity, we keep the same notation for the right inverses sincethey are defined on orthogonal spaces.

PROPOSITION 12.3 Assume that δ′ < −n/2 and α ∈ (0, 1) are fixed. There existssome constant c > 0, and, for all t0 ∈ R, there exists an operator

Gt0 : C0,α

δ′([t0,+∞) × S

n−1; 0 × 1) −→ C2,α

δ′([t0,+∞) × S

n−1; 0 × 1)such that, for all (g, 0) ∈ C0,α

δ′ ([t0,+∞) × Sn−1; 0 × 1), if for all t > t0, g(t, ·)

is constant on Sn−1, then ( f, 0) = Gt0(g, 0) is the unique solution of

(12.5) LH ( f, 0) = (g, 0)

in [t0,+∞) × Sn−1, which belongs to the space C2,α

δ′ ([t0,+∞) × Sn−1; 0 × 1).

Furthermore, ‖( f, 0)‖2,α,δ′ ≤ c‖(g, 0)‖0,α,δ′ .


PROOF: The existence of f is easy. Indeed, from Section 10, we know anexplicit solution of the homogeneous problem LH ( f0, 0) = 0. This solution corre-sponds to the Jacobi field �d and is given by

f0(t) := (cosh(nt))−1/2 .

Hence, we can define a solution of (12.5) by

f (t) := f0(t)∫ +∞

t( f0(ζ ))−2

∫ +∞

ζ

f0(ξ)g(ξ)dξ dζ .

It is a simple exercise to show that this solution is well defined and that the relevantestimate is satisfied, since we have chosen δ ′ < −n/2. Uniqueness follows at oncefrom the fact that the indicial roots of LH , restricted to V0, are equal to ±n/2. �

The final result of this section evaluates the norm of the Poisson operator asso-ciated to the operator L0.

PROPOSITION 12.4 There exists c > 0 such that for all W ∈ C2,α(Sn−1; 0 × 1),there exists a unique U ∈ C2,α

2−n2

([0,+∞) × Sn−1; 0 × 1) solution of

(12.6)

{L0U = 0 in (0,+∞) × S

n−1

U = W on {0} × Sn−1 .

Furthermore, we have ‖U‖2,α, 2−n2

≤ c‖W‖2,α.

PROOF: The uniqueness of the solution follows from the result of Proposi-tion 11.1. We therefore concentrate our attention on the existence of the solutionas well as the derivation of the estimate.

Step 1. Let us assume, for the time being, that the boundary data W is orthog-onal to V1

ex in the L2 sense on Sn−1. We apply Proposition 11.2, which implies that

the operator L0 is injective over (0, t ′) × Sn−1. As a consequence, for all t ′ > 0 we

are able to solve L0U = 0 in (0, t ′) × Sn−1 with U = W on {0} × S

n−1 and U = 0on {t ′} × S

n−1.We claim that, if δ ∈ (− n

2 , n2 ) is fixed, there exists some constant c > 0 inde-

pendent of t ′ > 1 and W such that

sup(0,t ′)×Sn−1

|e−δtU | ≤ c supSn−1

|W | .

In order to prove this estimate, we argue by contradiction and assume that theclaim is not true. In this case there would exist sequences t ′

i > 1 and a sequenceWi orthogonal to V1

ex in the L2 sense that satisfy

supSn−1

|Wi | = 1 ,


and a sequence Ui of solutions of L0Ui = 0 in (0, t ′i ) × S

n−1 with Ui = Wi on{0} × S

n−1 and Ui = 0 on {t ′i } × S

n−1 such that

Ai := sup(0,t ′i )×Sn−1

|e−δtUi | −→ +∞ .

Recall that, for all t , Ui (t, ·) is orthogonal to V1ex in the L2 sense on S

n−1. Let usdenote by (ti , θi ) ∈ (0, t ′

i ) × Sn−1 a point where the above supremum is achieved.

We now distinguish a few cases according to the behavior of the sequence −ti ,which, up to a subsequence, can always be assumed to converge in t∗ ∈ (−∞, 0).Up to some subsequence, we may also assume that the sequence t ′

i − ti convergesto t∗ ∈ (0,+∞]. Here, we have implicitly used the fact that the sequences ti andt ′i − ti remain bounded away from 0. This last fact can be proven as in the proof of

Proposition 12.2.We now define the rescaled sequence

Ui (t, θ) := e−δti

AiUi (t + ti , θ) .

After the extraction of some subsequences, if this is necessary, we may assume thatthe sequence Ui converges to some nontrivial solution of

(12.7) L0U∞ = 0

in (t∗, t∗) × Sn−1, with boundary condition U∞ = 0, if either t∗ or t∗ is finite.

Furthermore,

(12.8) sup(t∗,t∗)×Sn−1

|e−δtU∞| = 1

and U∞(t, ·) is orthogonal to V1ex in the L2 sense on S

n−1. Now the proof followsexactly the arguments of the proof of Proposition 12.2. We decompose

U = U 0 +∑j≥1

(U k

ex + U kcoex

),

where U 0 ∈ V0, U kex ∈ Vk

ex, U kcoex ∈ Vk

coex, and U 1ex = 0.

The inspection of the indicial roots of L0 together with the fact that U∞ isbounded by eδt for some δ ∈ (−n/2, n/2) shows that U 0, U k

ex, and U kcoex are

bounded by a constant times en2 t at −∞ if t∗ = −∞ and are bounded by a constant

times e− n2 t at +∞ if t∗ = +∞. Applying Proposition 12.2, we conclude that,

independently of the value of t∗ and t∗, U 0 = 0, U kex = 0 and U k

coex = 0 for allk ≥ 1, contradicting (12.8).

Having ruled out every possibility, the proof of the claim is complete. We canpass to the limit t ′ → +∞ in a sequence of solutions defined on (0, t ′) × S

n−1 andobtain a solution of L0U = 0, in (0,+∞) × S

n−1, with U = W on {0} × Sn−1,

which satisfiessup

(0,+∞)×Sn−1|e−δtU | ≤ c sup

Sn−1|W |


for some constant c > 0. Finally, Schauder’s estimates [7] yield the relevant esti-mates for the derivatives of U . Observe that, though the result is so far true for allδ ∈ (−n/2, n/2), we will ultimately be interested in the case where δ = (2 − n)/2.

Step 2. Assume that the boundary data W belongs to V 1ex, the eigenspace of the

operator (�0,�1) corresponding to the eigenvalue n − 1. In this case, the partialdifferential equation L0U = 0 reduces to a finite number of coupled ordinarydifferential equations of the form

a − (n − 1)a − n2

4a + 2(n − 1)b = 0

b − (n − 1)b − (n − 4)2

4b + 2a = 0

with boundary data a(0) = a0 and b(0) = b0. Then, provided �0φ = −(n − 1)φ,U = (aφ, bdφ) will be a solution of L0U = 0 in (0,+∞) × S

n−1 and U (0, ·) =(a0φ, b0dφ) on {0} × S

n−1.The above system can be solved explicitly, and we find{

a = (n − 1)αe− n+22 t + βe

2−n2 t

b = −αe− n+22 t + βe

2−n2 t ,

where nα = a0 − b0 and nβ = (n − 1)b0 + a0. Hence, we get

‖U‖2,α, 2−n2

≤ c(|a0| + |b0|) ≤ c‖W‖2,α .

This completes the proof of the result. �

13 Structure of the Mean Curvature Operator

In this section, we study the structure of the mean curvature operator for anysubmanifold that is close to a hyperbola. More precisely, any n-dimensional sub-manifold close enough to a hyperbola can be described, at least locally, as a normalgraph over a piece of the hyperbola. We would like to derive the equation whichensures that this normal graph produces a minimal submanifold. To do so, wecompute the first fundamental form of a normal graph over the hyperbola and goback to the variational definition of a minimal submanifold to derive the nonlinearequation.

Hence, we consider a (small) normal perturbation of the hyperbola, which isassumed to be parametrized by X . Recall that, with our notation,

X = eis

(sin(ns))1/n�,

and that a (small) normal vector field N can be taken to be

N = iei(1−n)s f � + ieis T .


We define Y := X + N . The tangent vectors to the submanifold parametrized byY are given by

∂t Y =(

−(cosh(nt))1/n +(

(n − 1)

cosh(nt)f + i∂t f

))ei(1−n)s�

− eis(

1

cosh(nt)T − i∂t T

),

and, for all j = 1, . . . , n − 1,

∂j Y = (cosh(nt))1/neis�j + iei(1−n)s(∂θj f � + f �j ) + ieis∂θj T .

The exact value of the coefficients of the first fundamental form of the submanifoldparametrized by Y is not needed for the forthcoming analysis. Just observe that thefirst fundamental form of the submanifold parametrized by Y can be expanded as

IY = IX + (cosh(nt))(1−n)/n L( f, T ) + Q( f, T ) ,

where IX denotes the first fundamental form of the hyperbola (in the variables(t, θ)), namely,

IX = (cosh(nt))2/n(dt ⊗ dt + dθi ⊗ dθj ) ,

and where L is linear and Q is quadratic in the variables f , T , ∂t f , ∂t T , εj f , and∇τ

εjT . The key observation is that the operators L and Q have coefficients that

depend on t but are bounded and have bounded partial derivatives with respect to t .Once we have obtained the structure of the first fundamental form, it is a simple

exercise to obtain the structure of the volume functional and then the structure ofthe corresponding Euler-Lagrange equation. We leave the details to the reader andrefer to [4] for further details on the derivation of a similar operator in a slightlydifferent context. Identifying the C2,α normal vector field N with U ∈ C2,α(R ×S

n−1, 0 × 1), as usual, we conclude that the submanifold parametrized by Y isminimal if and only if U is a solution of

L H U = (cosh(nt))−(1+n)/n Q2

(U

(cosh(nt))1/n

)

+ (cosh(nt))−1/n Q3

(U

(cosh(nt))1/n

),

where L H is the Jacobi operator, Q2 is homogeneous of degree 2, and Q3 col-lects all the higher-order terms. Here the important observation is the fact thatthe coefficients of L and Q are bounded functions of t implies that the coeffi-cients of the Taylor expansion of Q2 and Q3 are also bounded functions of t .More precisely, there exists a constant c > 0 such that, for all t ∈ R and allW1, W2 ∈ C2,α([t, t + 1] × S

n−1, 0 × 1),

(13.1) ‖Q2(W1) − Q2(W2)‖0,α ≤ c(‖W2‖2,α + ‖W1‖2,α

)‖W2 − W1‖2,α ,

and, provided ‖W2‖2,α ≤ 1 and ‖W1‖2,α ≤ 1, we also have

(13.2) ‖Q3(W1) − Q3(W2)‖0,α ≤ c(‖W2‖2

2,α + ‖W1‖22,α

)‖W2 − W1‖2,α .


Here all norms are evaluated over [t, t + 1] × Sn−1. The key point here is that the

constant c does not depend on t .

14 Minimal Submanifolds Close to a Hyperbola

If we conjugate the above nonlinear operator, as we have done with the operatorL H , we conclude that the submanifold parametrized by

Y = X + (cosh(nt))(2−n)/2n N

is minimal if and only if U (which is associated to the normal vector field N ) is asolution of

LH U = (cosh(nt))−1/2 Q2

(U

(cosh(nt))1/2

)+ (cosh(nt))1/2 Q3

(U

(cosh(nt))1/2

).

We assume that t∗ < 0 is fixed (and, when n = 3, we further assume that t∗ < −t0).We modify the normal bundle when (t, θ) ∈ [t∗, t∗ + 2] × S

n−1 so that it is nowgiven by the “vertical plane” {i x : x ∈ R

n} when (t, θ) ∈ [t∗, t∗ + 1] × Sn−1.

For example, this can be achieved by considering, instead of the normal bundlespanned by

N0 = iei(1−n)s� and ∀ j = 1, . . . , n N j = ieis �j

|�j | ,

the vector bundle spanned by

N0 := i� and ∀ j = 1, . . . , n Nj := i�j

|�j |in [t∗, t∗ + 1] × S

n−1, and then by interpolating smoothly between these vectorspaces. As explained in [16], this modification slightly alters the equation we haveto solve into

LH U = (cosh(nt))−2 LU + (cosh(nt))−1/2 Q2

(U

(cosh(nt))1/2

)

+ (cosh(nt))1/2 Q3

(U

(cosh(nt))1/2

),

(14.1)

where the linear operator L has coefficients that are bounded and supported in[t∗, t∗ + 2] × S

n−1 and where the nonlinear operators Q2 and Q3 satisfy (13.1) and(13.2) (with Q j replaced by Q j ). The coefficient (cosh(nt))−2 in front of L comesfrom the fact that the scalar product of N0 (respectively, Nj ) with N0 (respectively,Nj ) is equal to 1 + O((cosh(nt))−2) and hence, the change of bundles induces alinear operator whose coefficients are bounded by a constant times ((cosh(nt))−2).

Given W ∈ C2,α(Sn−1; 0 × 1), which is orthogonal to V0 in the L2 sense,we want to find a solution U of (14.1) in (t∗,+∞) × S

n−1, with boundary dataU − W ∈ V0 on {t∗} × S

n−1. We shall assume that

‖W‖2,α ≤ κe5n2 t∗


for some fixed constant κ > 0.To begin with, we make use of the result of Proposition 12.4 to solve

(14.2)

{L0U0 = 0 in (t∗,+∞) × S

n−1

U0 = W on {t∗} × Sn−1 .

It also follows from Proposition 12.4 that

(14.3) ‖U0‖2,α, 2−n2

≤ cen−2

2 t∗‖W‖2,α .

It remains to solve

(14.4) LHU = NW (U ) ,

where the nonlinear operator NW is given by

NW (U ) := (cosh(nt))−2 L(U + U0) − LH U0

+ (cosh(nt))−1/2 Q2

(U + U0

(cosh(nt))1/2

)

+ (cosh(nt))1/2 Q3

(U + U0

(cosh(nt))1/2

).

Making use of the result of Proposition 12.2 and Proposition 12.3, we may rewritethe previous equation as

U = Gt∗NW (U ) .

Finally, in order to solve this equation, we use a fixed-point theorem for contractionmapping. This is the content of the following analysis.

Step 1. To begin with, we use (14.3) and the properties of L to estimate

‖(cosh(nt))−2 LU0‖0,α,δ ≤ ce(2n−δ)t∗‖W‖2,α .

This first estimate does not require any assumption on the weight δ. Now, observethat

LH U0 = (LH − L0)U0 .

This, together with the expression of LH and L0, yields

‖LH U0‖0,α,δ ≤ c(e(2n−δ)t∗ + e

n−22 t∗)‖W‖2,α ,

provided δ ≥ (2 − n)/2.Now, using (13.1), we can estimate∥∥∥∥(cosh(nt))−1/2 Q2

(U0

(cosh(nt))1/2

)∥∥∥∥0,α,δ

≤ c(e( 3n

2 −δ)t∗ + e(n−2)t∗)‖W‖22,α ,

provided δ ≥ (4 − 5n)/2. Finally, using (13.2), we get∥∥∥∥(cosh(nt))1/2 Q3

(U0

(cosh(nt))1/2

)∥∥∥∥0,α,δ

≤ c(e(n−δ)t∗ + e

32 (n−2)t∗)‖W‖3

2,α ,

provided t∗ is chosen negative enough and δ ≥ (6 − 5n)/2.


Step 2. Assume that δ ∈ ( 2−n2 , n−2

2 ). We define P0 to be the orthogonal pro-jection onto V0. We can now use the result of Proposition 12.2, together with theestimates obtained in step 1, with δ = δ, to conclude that

(14.5) ‖Gt∗(I − P0)NW (0)‖2,α,δ ≤ cen−2

2 t∗‖W‖2,α .

Step 3. It remains to estimate Gt∗ P0NW (0). To begin with, observe that

P0LH U0 = 0 ,

since U0 is orthogonal to V0. Next, we assume that δ ∈ ( 2−n2 , n−2

2 ), and we use theestimates of step 1 with δ = δ + 1 − n. Observe that δ < − n

2 ; hence, we can applythe result of Proposition 12.3 together with the obvious inequality

‖V ‖2,α,δ ≤ e(δ′−δ)t∗‖V ‖2,α,δ′ ,

which holds when δ′ ≤ δ, to conclude that

‖Gt∗ P0NW (0)‖2,α,δ ≤ cen−2

2 t∗‖W‖2,α .

Collecting this last estimate and (14.5), we conclude that there exists a constantc∗ > 0, which does not depend on t∗, such that

(14.6) ‖Gt∗NW (0)‖2,α,δ ≤ c∗en−2

2 t∗‖W‖2,α ,

provided t∗ is chosen negative enough.

Step 4. Using similar arguments, we can prove that

‖Gt∗(NW (U2) − NW (U1))‖2,α,δ ≤ 1

2‖U2 − U1‖2,α,δ ,

provided t∗ is chosen negative enough and 2−n2 < δ < n

6 . Observe the extra restric-tion that is needed on the weight parameter δ. The details are left to the reader.

Given steps 1 through 4, standard arguments now imply that the mapping

U −→ Gt∗NW (U )

has a unique fixed point in the ball of radius 2c∗en−2

2 t∗‖W‖2,α in C2,αδ (R × S

n−1; 0 × 1), provided t∗ is chosen negative enough. Furthermore, reducing t∗ if thisis necessary, we can assume that the solution U depends smoothly on the data W .

We summarize what we have obtained so far and translate the result in a moregeometric framework.

Given β ∈ (0,+∞) and ε ∈ (0, 1), we define

rε := ε2/(3n) ,

and we define sε ∈ (0, π/n) by

rε = (nβε)1/n cos sε

(sin(nsε))1/n.


Next we define tε ∈ R by

e−ntε = sin(nsε)

1 − cos(nsε).

Observe that, as ε tends to 0, keeping β bounded and bounded away from 0, tεtends to −∞. More precisely, we have

tε = 1

3nlog ε + O(1)

as ε tends to 0.Given � ∈ C2,α(Sn−1; R

n), which is orthogonal to � in the L2 sense on Sn−1

and which satisfies‖�‖2,α ≤ κεrε ,

we identify � with V ∈ C2,α(Sn−1; 0 × 1). Next, we define

W := (nβε)−1/n(cosh(ntε))(n−2)/(2n)V .

Provided ε is chosen small enough (say ε ∈ (0, ε0)), the previous analysis can beapplied to t∗ = tε and W and yields the existence of U + U0 ∈ C2,α

δ ([tε,+∞) ×S

n−1; 0 × 1), where U is the solution of (14.2) and U0 is the solution of (14.4).In turn, we now identify U + U0 with a normal vector field N . We conclude thatthe n-diemensional submanifold that is parametrized by

Y = (nβε)1/n(X + (cosh(nt))(2−n)/(2n)N)

for (t, θ) ∈ [tε,+∞)×Sn−1 is a minimal submanifold that is close to the hyperbola

and whose boundary is, in some sense, parametrized by �.Now, we give a more precise description of this submanifold close to its bound-

ary. To this aim, we define, as in Section 6,

r := (nβε)1/n cos s(sin(ns))1/n

.

Thanks to (6.1), we find, with little work, that the above-defined minimal subman-ifold can be parametrized, in some neighborhood of its boundary, as the graph of

(r, θ) −→ r� + i(εβr1−n� + F + F ,β

),

where F is the unique, smooth harmonic extension of the boundary data � in Brε

and where F ,β satisfies

(14.7) ‖F ,β(rε, ·)‖2,α + ‖rε∂r F ,β(rε, ·)‖1,α ≤ cεrε

for some constant c > 0 that does not depend on κ or on β, provided this latter isassumed to belong to some fixed compact subset of (0,+∞). The constant c thatappears in this estimate is essentially given by the term O(ε3r1−3n) in the expansion(6.1). Here, all norms are computed over S

n−1. Observe that we can assume thatF ,β depends smoothly on the data � and β provided we restrict ε in a smallerrange.


Finally, given R ∈ O(n) we consider the image of the above-defined family ofminimal submanifolds by

x + iy ∈ Cn −→ x + iRy ∈ C

n .

We denote by Hε,R(β,�) this submanifold.For further use it will be convenient to make the following definition:

DEFINITION 14.1 For all � ∈ C2,α(Sn−1; Rn), we define Pint(�) to be equal to

∂r F |∂ B1 where F is the unique smooth harmonic extension in B1(0) of the bound-ary data � on ∂ B1(0).

15 Mapping Properties of the Laplace Operator in Rn

In this section we study the mapping properties of the Laplace operator in Rn

acting on some weighted Hölder spaces.To define the spaces we are going to work with, we assume that x1, . . . , xN ∈

Rn are given, and we choose ρ0 > 0 such that, for all j �= j ′, we have

B2ρ0(x j ) ∩ B2ρ0(x j ′) = ∅

and alsoB2ρ0(x j ) ⊂ B1/ρ0(0) .

The weighted spaces we work with are defined as follows:

DEFINITION 15.1 Assume that ν, µ ∈ R and α ∈ (0, 1) are fixed. The spaceCk,α

ν,µ(Rn − {x1, . . . , xN }; Rn) is defined to be the space of

F ∈ Ck,α(Rn − {x1, . . . , xN }; Rn)

for which the following norm is finite:

‖F‖k,α,ν,µ :=∑

j

supr∈(0,ρ0)

r−ν‖F(r · +x j )‖k,α,B2(0)−B1(0)

+ ‖F‖k,α,B2/ρ0 (0)−⋃j Bρ0 (x j ) + sup

r≥1/ρ0

r−µ‖F(r ·)‖k,α,B2(0)−B1(0) ,

where ‖ · ‖k,α,� is the usual Hölder norm in .

Roughly speaking, we ask that the components of F be functions that, for all j ,are bounded by a constant times |x−x j |ν near x j and that are bounded by a constanttimes |x |µ at ∞. If ⊂ R

n − {x1, . . . , xN }, we define the space Ck,αν,µ( ; R

n) tobe the space of restrictions of functions of Ck,α

ν,µ(Rn − {x1, . . . , xN }; Rn) to . This

space is endowed with the induced norm.This definition being understood, we prove the following:

PROPOSITION 15.2 Assume that 2 − n < ν,µ < 0 are fixed. Then

� : C2,αν,µ(Rn − {x1, . . . , xN }; R

n) −→ C0,αν−2,µ−2(R

n − {x1, . . . , xN }; Rn)

is an isomorphism.


PROOF: Since � acts diagonally on the components of F , it is enough to provethe result for scalar functions. Let χ be a cutoff function that is identically equal to1 in R

n − B2/ρ0(0) and equal to 0 in B1/ρ0(0). Observe that, if g ∈ C0,αν−2,µ−2(R

n −{x1, . . . , xN }), then (1 − χ)g ∈ L1(Rn). Hence, there exists a function f1 that is a(weak) solution of � f1 = (1 − χ)g in R

n . Regularity theory then implies that f1is a solution of this equation away from the points x j and that there exists c > 0(depending on ρ0) such that

‖ f1‖2,α,B2/ρ0 (0)−⋃j Bρ0 (x j ) ≤ c‖g‖0,α,ν−2,µ−2 .

Now, observe that the function x → |x − x j |ν can be used as a barrier to prove thatf1 is bounded by a constant times ‖g‖0,α,ν−2,µ−2|x − x j |ν in each Bρ0(x j ). Thisfollows immediately from the fact that

�r ν = ν(n − 2 + ν)r ν−2

and ν(n − 2 + ν) is negative exactly when ν ∈ (2 − n, 0).In R

n − B1/ρ0(0), f1 is bounded by a constant times ‖g‖0,α,ν−2,µ−2|x |2−n . Thisfact follows immediately from Green’s representation formula for f1 together withthe fact that (1 − χ)g has compact support. The estimates for the derivatives of f1follow from (rescaled) Schauder’s estimates and are left to the reader. We concludethat

‖ f1‖2,α,ν,2−n ≤ c‖g‖0,α,ν−2,µ−2 .

Since the function χg is bounded by a constant times |x |µ−2, the function x →|x |µ can be used as a barrier to prove that there exists a solution f2 of � f2 = χgthat is bounded by a constant times ‖g‖0,α,ν−2,µ−2|x − x j |µ. Moreover, since thefunction χg has support away from the origin, the function f2 is in fact C2,α, andwe conclude that

‖ f2‖2,α,ν,µ ≤ c‖g‖0,α,ν−2,µ−2 .

The surjectivity property then follows at once. The injectivity property simplyfollows from the fact that the singularities of any solution of � f = 0 that belongsto C2,α

ν,µ(Rn − {x1, . . . , xN }) are removable. Therefore, the function f is a boundedharmonic function and hence is equal to 0. �

Using similar arguments (or the previous construction together with the maxi-mum principle), one can also show the following:

PROPOSITION 15.3 Assume that 2 − n < ν,µ < 0 are fixed. Then, for all ρ ∈(0, ρ0) the map

� :(C2,α

ν,µ

(R

n −⋃

j

Bρ(x j )

); R

n)

0−→ C0,α

ν−2,µ−2

(R

n −⋃

j

Bρ(x j ); Rn)

is an isomorphism. Moreover, there exists c > 0 such that, for all ρ ∈ (0, ρ0), thenorm of its inverse Gρ is bounded by c.

Here the subscript 0 refers to the fact that the functions have 0 boundary dataon each ∂ Bρ(x j ).


16 Expansion of Green’s Function

Assuming that x1, . . . , xN ∈ Rn , α1, . . . , αN ∈ R, A0 ∈ Mn(R), and

R1, . . . ,RN ∈ O(n) are given, we define

G(x) :=N∑

j=1

αjRj

(x − x j

|x − x j |n)

+ A0x .

The asymptotic expansion of G near the point x j0 is given by

G(x) = αj0r1−nRj0� −

( ∑j �= j0

αjRj

(x j − x j0

|x j − x j0 |n)

− A0x j0

)

+ r(A0� +

∑j �= j0

αj

|x j − x j0 |nRj (� − n(� · ξj, j0)ξj, j0)

)+ O(r2) ,

where we have definedr� := x − x j0 ,

and, as in the introduction,

ξj j ′ := x j − x j ′

|x j − x j ′ |for all j �= j ′. We further define, as in the introduction,

γj j ′ := 1

|x j − x j ′ |n∫

Sn−1

(Rj� · Rj ′� − n(� · Rjξj, j ′)(� · Rj ′ξj, j ′))dθ

for all j �= j ′ and

λj := −∫

Sn−1

A0� · Rj� dθ .

This being understood, we find that the asymptotic expansion of G near x j0 canbe refined into

G(x) =(

αj0r1−n + 1

ωn

( ∑j �= j0

αjγj j0 − λj0

)r)Rj0�

−( ∑

j �= j0

αjRj

(x j − x j0

|x j − x j0 |n)

− A0x j0

)+ O(r2) + O⊥(r) ,

(16.1)

where O⊥(r) is orthogonal to Rj0� in the L2 sense on Sn−1 and ωn := |Sn−1|.

17 Minimal Graphs

Consider a “vertical” graph over the x-space in Cn , namely,

⊂ Rn � x −→ x + i F(x) ∈ C

n .


The first fundamental form of this submanifold is then given by

I =∑

j

dx j ⊗ dx j +∑j, j ′

∂x j F · ∂x ′jF dx j ⊗ dx j ′ .

It is an exercise to derive the equation which ensures that the graph of F is minimal.We do not need the exact expression of this equation but only its structure

�F + div(Q3(∇F)

) = 0 ,

where the nonlinear operator

M −→ Q3(M)

satisfies the following: There exists c > 0 and c0 > 0 such that, for all M1, M2 ∈Mn(R

n),

(17.1)∣∣Q3(M2) − Q3(M1)

∣∣ ≤ c(|M2|2 + |M1|2

)|M2 − M1| ,provided |Mj | ≤ c0. Here we have used the natural identification of ∇F(x) withan n × n matrix.

We keep the notation and definitions of the previous section for the function Gand further assume that κ > 1 and ε ∈ (0, 1) are given. For all �1, . . . , �N ∈C2,α(Sn−1; R

n) satisfying‖�j‖2,α ≤ κεrε ,

we would like to solve

�(F + εG) + div(Q3(∇(F + εG))

) = 0 in Rn −

⋃j

Brε(x j ) ,

with boundary data given by

F = Rj�j on ∂ Brε(x j ) .

We define the harmonic function F0 = ∑j Fj where, for each j , the function

Fj is the solution of {�Fj = 0 in R

n − Brε(x j )

Fj = Rj�j on ∂ Brε(x j ) .

Using barrier arguments, we prove the following lemma:

LEMMA 17.1 There exists a constant c > 0 such that

‖F0‖2,α,2−n,2−n ≤ crn−2ε sup

j‖�j‖2,α .

PROOF: Since the Laplacian acts diagonally on the components of Fj , it isenough to prove the result for � f = 0 in R

n − Brε(0) and f = h on ∂ Brε

(0).Observe that the function

x −→ |x |2−n


can be used as a barrier to show the pointwise estimate

| f (x)| ≤ r n−2ε

(sup

∂ Brε (0)

|h|)|x |2−n .

The estimates for the other derivatives follow from Schauder’s estimates. �

We setF := εG + F0 ,

and we prove the following:

LEMMA 17.2 Assume that 2 − n < ν,µ < 0 are fixed. There exists a constantc > 0 (independent of κ) and a constant ε0 > 0 (dependent on κ) such that, for allε ∈ (0, ε0), we have∥∥�F + div(Q3(∇ F))

∥∥0,α,ν−2,µ−2 ≤ cεr1−ν

ε ,

provided the coefficients αj remain in a fixed compact subset of (0,+∞).

PROOF: Observe that �F = 0 in Rn − ⋃

j Brε(x j ). It remains to estimate

div(Q3(∇ F)). To do so, first observe that we have div(Q3(∇ F)) = 0 for F(x) :=εA0x , since ∇ F is constant. Using this and Lemma 17.1, together with (17.1), wecan evaluate ∥∥div

(Q3(∇ F) − Q3(∇ F)

)∥∥2,α,ν−2,µ−2 ≤ cεr1−ν

ε

provided ε is chosen small enough. The value of the constant c is essentially deter-mined by the range in which the constants α j are chosen, all other terms being ofsmaller order if ε is chosen small enough. This ends the proof of the lemma. �

Now we want to perturb the graph of F in order to obtain a minimal submani-fold. The equation we have to solve now reads

(17.2)

{�(F + F) + div

(Q3(∇(F + F))

) = 0 in Rn − ⋃

j Brε(x j )

F = 0 on ∂ Brε(x j ) .

Making use of Proposition 15.3, we can rewrite this equation as

F = −GrεMε, (F) ,

where we have defined

Mε, (F) := �F + div(Q3(∇(F + F))

).

Again we use a fixed-point argument for contraction mapping to produce a solutionof (17.2). This is the content of the following analysis.

It follows from Lemma 17.2 that there exists c0 > 0 (independent of κ) andε0 > 0 (dependent on κ) such that, for all ε ∈ (0, ε0),

‖GrεMε, (0)‖2,α,ν,µ ≤ c0εr1−ν

ε


and

‖Grε(Mε, (F2) − Mε, (F1))‖2,α,ν,µ ≤ 1

2‖F2 − F1‖2,α,ν,µ

for all F1, F2 ∈ (C2,αν,µ(Rn − ⋃

j Br∗(x j )); Rn)0 such that ‖Fi‖2,α,ν,µ ≤ 2c0εr1−ν

ε .Everything here is uniform in the αj provided these are chosen in some fixed com-pact subset of (0,+∞). In particular, the nonlinear mapping F −→ −Gr∗Mε,

has a unique fixed point in the ball of radius 2c0ε3 in the space(

C2,αν,µ

(R

n −⋃

j

Brε(x j )

); R

n)

0.

We give a summary of what we have obtained in a more geometric setting. Wehave assumed that x1, . . . , xN ∈ R

n , A0 ∈ Mn(R), and R1, . . . ,RN ∈ O(n) arefixed. Given κ > 1, there exists ε0 > 0 such that for all αj in some fixed compactsubset of (0,∞) for all ε ∈ (0, ε0), and for all �j ∈ C2,α(Sn−1; R

n), which we nowassume to be orthogonal to Rj� in the L2 sense on S

n−1 and which satisfy

‖�j‖2,α ≤ κεrε ,

we have obtained a minimal n-dimensional submanifold that is close to the n-plane

x −→ x + iεA0x .

Furthermore, this submanifold has N boundaries and in some neighborhood ofeach of its boundaries, it can be parametrized (up to some translation) as the graphof

(r, θ) −→(

εαj0r1−n + ε

ωn

( ∑j �= j0

γj0 jαj − λj0

)r)Rj0� + J j0

+ Jj0 + J ⊥j0 ,

where we have setr� := x − x j0

for (r,�) ∈ [rε, ρ0] × Sn−1. Here J j is the unique harmonic extension in R −

Brε(x j ) of the boundary data Rj�j on ∂ Brε

(x j ), which tends to 0 at ∞. Moreover,Jj0 and J ⊥

j0 depend nonlinearly on the data �j and αj , and they satisfy

(17.3) ‖Jj0(rε, ·)‖2,α + ‖rε∂r Jj0(rε, ·)‖1,α ≤ cεr2ε

and

(17.4) ‖J ⊥j0 (rε, ·)‖2,α + ‖rε∂r J ⊥

j0 (rε, ·)‖1,α ≤ cεrε

for some constant c > 0 that does not depend on κ , provided ε ∈ (0, ε0) and theαj are chosen in some fixed compact subset of (0 + ∞). In addition, J ⊥

j0 (r, ·) isorthogonal to Rj0� in the L2 sense on S

n−1, while Jj0(r, ·) is collinear to Rj0�.Observe that, provided ε is chosen small enough, we can assume that the mappingsJ ⊥

j0 and Jj0 depend smoothly on the data �j and αj . We denote by �ε((αj )j , (�j )j )

this submanifold.For further use it will be convenient to have the following definition:


DEFINITION 17.3 For all �j ∈ C2,α(Sn−1; Rn), we define Pext(�) to be equal to

∂r J |∂ B1(0) where J is the unique harmonic extension in R−B1(0) of the boundarydata �j on ∂ B1(0) that tends to 0 at ∞.

18 Gluing Procedure

Assume that x1, . . . , xN ∈ Rn , A0 ∈ Mn(R), and R1, . . . ,RN ∈ O(n) are

fixed in such a way that (H1), (H2), and (H3) hold. By assumption, we can choose(α∗

1 , . . . α∗N ) ∈ R

N+ such that ∑j

γj j ′α∗j ′ = λj .

We set

β∗j = α∗

j .

Finally, we fix κ > 0 large enough. By the previous analysis there exists ε0 > 0such that, for all ε ∈ (0, ε0) and for all αj , βj in some fixed compact subsetof (0,∞) that contains all the α∗

j , for all �j , �j ∈ C2,α(Sn−1; Rn) that are or-

thogonal to � in the L2 sense on Sn−1, we can define, as in the previous sec-

tion, the minimal submanifold �ε((αj )j , (�j )j ). We can also define a minimaln-submanifold Hε,Rj (βj , �j ), which, once translated by the vector x j , will be de-noted by Hε(βj , �j ).

Our aim is now to find �j , �j , αj , and βj in such a way that

Mε := �ε((αj )j , (�j )j )

N⋃j ′=1

Hε(βj ′,�j ′)

is a C1 submanifold.Writing that the boundaries of these submanifolds coincide yields the following

system of equations:

�j − �j = O(εrε)

and

ε(βj0 − αj0)r1−nε + ε

ωn

∑j �= j0

γj0 j (αj − α∗j )rε = O(

εr2ε

),

where the first equation corresponds to the projection of the boundary data overthe space of functions orthogonal to � in the L2 sense on S

n−1 and where the sec-ond equation corresponds to the orthogonal projection over the space of functionsspanned by �.

Similarly, writing that the conormals at the boundaries coincide yields the fol-lowing system of equations:

Pext(�j ) − Pint(�j ) = O(εrε)


and(1 − n)ε(βj0 − αj0)r

1−nε + ε

ωn

∑j �= j0

γj0 j (αj − α∗j )rε = O(

εr2ε

),

where the first equation corresponds to the projection over the space of functionsorthogonal to � in the L2 sense on S

n−1 and where the second equation correspondsto the orthogonal projection over the space of functions spanned by �. The O arenonlinear functions of the data �j , �j , αj , and βj . The estimates follow from(14.7), (17.3), and (17.4).

We will now use the following:

LEMMA 18.1 The mapping

Pext − Pint : C2,α(Sn−1; Rn) −→ C1,α(Sn−1; R

n)

is an isomorphism.

PROOF: Observe that both Pint and Pext are self-adjoint first-order pseudodif-ferential operators that are elliptic with principal symbols |ξ | and −|ξ |, respec-tively; hence the difference is also elliptic and semibounded. This means thatPext −Pint has discrete spectrum, and thus we only need to prove that it is injective.The invertibility in Hölder spaces then follows by standard regularity theory.

To prove this, we argue by contradiction. Assume that Pext−Pint is not injective.Then, there would exist a function � ∈ C2,α(Sn−1; R

n) for which

(Pext − Pint)(�) = 0 .

We may extend the Dirichlet data � to a harmonic function F on the B1(0) andalso on R

n − B1(0). In addition, F tends to 0 at ∞. By construction F is C1 andhence is C∞ and tends to 0 at ∞. Thus F ≡ 0. This completes the proof of theinjectivity of Pext − Pint. �

We setFα := (

(C2,α(Sn−1; Rn))2 × R

2)N

endowed with the product norm. By using this result, the previous system of equa-tions reduces to(

�j , �j , ε(αj − α∗j ), ε(βj − β∗

j ))

j = C(�j , �j , ε(αj − α∗

j ), ε(βj − β∗j )

)j ,

where the nonlinear mapping C satisfies

(18.1)∥∥C

(�j , �j , ε(αj − α∗

j ), ε(βj − β∗j )

)j

∥∥F ≤ cεrε

for some constant that does not depend on κ , provided ε is chosen small enough,say ε ∈ (0, ε0).

We denote by Bακ the ball of radius κεrε in Fα. It follows from our previous

analysis that, for fixed κ > 0 large enough, the mapping C is well-defined inBα

κ provided we choose the parameter ε to be small enough. The value of κ isdirectly related to the constant that appears in (18.1). To conclude, we want touse Schauder’s fixed-point theorem, which will ensure the existence of at least one


fixed point of C in Bακ . This zero of C produces a C1,α minimal n-submanifold Mε.

It is then a simple exercise to see, thanks to regularity theory, that Mε is in fact aC∞ minimal hypersurface with N + 1 ends.

However, since C is not compact, it is not possible to apply directly Schauder’stheorem. This is the reason why, as in [4], we introduce a family of smoothingoperators Dq , for all q > 1, which satisfy for fixed 0 < α ′ < α < 1,

‖Dq�‖2,α′ ≤ c0‖�‖2,α , ‖Dq�‖2,α ≤ c0qα−α′‖�‖2,α′ ,

and

(18.2) ‖� − Dq�‖2,α′ ≤ c0qα′−α‖�‖2,α

for some constant c0 > 0 that does not depend on q > 1. Here all norms areunderstood over S

n−1. The existence of such smoothing operators is available in[1, prop. 1.6, p. 97]. To keep the notation short, we use the same notation for thesmoothing operator defined on Fα and acting on all function spaces.

Now we fix κ > 0 large enough. For all q > 1, we may apply Schauder’sfixed-point theorem to DqC to obtain the existence of a fixed point Pq of DqC inBα

κ , provided ε is chosen small enough, say ε ∈ (0, ε0).Since Pq has norm bounded uniformly in q, we may extract a sequence q j →

+∞ such that Pqj converges in Fα′for some fixed α′ < α. Thanks to the continuity

of C (with respect to the C2,α′and C1,α′

topology) and also to (18.2), the limit ofthis sequence is a fixed point of the mapping C and hence produces a zero of C forall ε ∈ (0, ε0). This completes our proof of the theorem.

Bibliography[1] Alinhac, S.; Gérard, P. Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs

Actuels. InterEditions, Paris; Éditions du Centre National de la Recherche Scientifique (CNRS),

Meudon, 1991.

[2] Bott, R.; Tu, L. W. Differential forms in algebraic topology. Graduate Texts in Mathematics,

82. Springer, New York–Berlin, 1982.

[3] Butscher, A. Regularizing a singular special Lagrangian variety. Comm. Anal. Geom., in press.

Available online at http://arxiv.org/abs/math.DG/0110053.

[4] Fakhi, S.; Pacard, F. Existence result for minimal hypersurfaces with a prescribed finite number

of planar ends. Manuscripta Math. 103 (2000), no. 4, 465–512.

[5] Folland, G. B. Harmonic analysis of the de Rham complex on the sphere. J. Reine Angew. Math.398 (1989), 130–143.

[6] Gallot, S.; Meyer, D. Opérateur de courbure et laplacien des formes différentielles d’une variété

riemannienne. J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284.

[7] Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order. Grundlehren

der Mathematischen Wissenschaften, 224. Springer, Berlin–New York, 1977.

[8] Harvey, F. R. Spinors and calibrations. Perspectives in Mathematics, 9. Academic Press,

Boston, 1990.

[9] Harvey, R.; Lawson, H. B., Jr. Calibrated geometries. Acta Math. 148 (1982), 47–157.


[10] Isenberg, J.; Mazzeo, R.; Pollack, D. Gluing and wormholes for the Einstein constraint equa-

tions. Comm. Math. Phys., in press. Available online at

http://front.math.ucdavis.edu/gr-qc/0109045.

[11] Joyce, D. On counting special Lagrangian homology 3-spheres. Available online at

http://arxiv.org/list/math.DG/9907.

[12] Lawlor, G. The angle criterion. Invent. Math. 95 (1989), no. 2, 437–446.

[13] Lawson, H. B., Jr. Lectures on minimal submanifolds. Vol. I. Second edition. Mathematics

Lecture Series, 9. Publish or Perish, Wilmington, Del., 1980.

[14] Lawson, H. B., Jr.; Michelsohn, M.-L. Spin geometry. Princeton Mathematical Series, 38.

Princeton University Press, Princeton, N.J., 1989.

[15] Lee, Y. I. Embedded special Lagrangian submanifolds in Calabi-Yau manifolds. Preprint. Avail-

able online at

http://math.cts.nthu.edu.tw/Mathematics/english/preprints/author.html.[16] Mazzeo, R.; Pacard, F. Constant mean curvature surfaces with Delaunay ends. Comm. Anal.

Geom. 9 (2001), no. 1, 169–237.

[17] Paquet, L. Méthode de séparation des variables et calcul de spectre d’opérateurs sur les formes

différentielles. C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 2, A107–A110.

[18] Ross, M. Complete minimal spheres and projective planes in Rn with simple ends. Math. Z.

201 (1989), no. 3, 375–380.

[19] Salur, S. Deformations of special Lagrangian submanifolds. Commun. Contemp. Math. 2(2000), no. 3, 365–372.

[20] Salur, S. A gluing theorem for special Lagrangian submanifolds. Available online at

http://arxiv.org/abs/math.DG/0108182.

[21] Salur, S. On constructing special Lagrangian submanifolds by gluing. Available online at

http://arxiv.org/pdf/math.DG/0201217.

[22] Strominger, A.; Yau, S. T.; Zaslow, E. Mirror symmetry is T -duality. Nucl. Phys. B. 479 (1996)

no. 1-2, 243–259. Available online at

http://miyako.math.metro-u.ac.jp/ masanori/QMSrefs.html.

CLAUDIO AREZZO FRANK PACARDUniversity of Parma Université Paris 12Department of Mathematics UFR des Sciences et TechnologieVia Massimo D’Azeglio 85/A Bât. P3 - 4ème étageI - 43100 Parma 61, avenue du Général de GaulleITALY 94010 Créteil CedexE-mail: claudio.arezzo@ FRANCE

unipr.it E-mail: [email protected]

Received April 2001.

Complete, Embedded, Minimal n-Dimensional Submanifolds in Cn CLAUDIO AREZZO Università di Parma AND...

Documents

Transcript of Complete, Embedded, Minimal n-Dimensional Submanifolds in Cn CLAUDIO AREZZO Università di Parma AND...