K. Funano - Eigenvalues of Laplacian and Multi-Way Isoperimetric Constants on Weighted Riemannian...

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EIGENVALUES OF LAPLACIAN AND MULTI-WAY

ISOPERIMETRIC CONSTANTS ON WEIGHTED

RIEMANNIAN MANIFOLDS

KEI FUNANO

Abstract. We investigate the distribution of eigenvalues of theweighted Laplacian on closed weighted Riemannian manifolds ofnonnegative Bakry-Emery Ricci curvature. We derive some uni-versal inequalities among eigenvalues of the weighted Laplacianon such manifolds. These inequalities are quantitative versions ofthe previous theorem by the author with Shioya. We also studysome geometric quantity, called multi-way isoperimetric constants,on such manifolds and obtain similar universal inequalities amongthem. Multi-way isoperimetric constants are generalizations of theCheeger constant. Extending and following the heat semigroup ar-gument by Ledoux and E. Milman, we extend the Buser-Ledouxresult to the k-th eigenvalue and the k-way isoperimetric constant.As a consequence the k-th eigenvalue of the weighted Laplacian andthe k-way isoperimetric constant are equivalent up to polynomialsof k on closed weighted manifolds of nonnegative Bakry-EmeryRicci curvature.

1. Introduction

1.1. Eigenvalues of the weighted Laplacian. Let (M,) be a pairof a Riemmanian manifold M and a Borel probability measure onM of the form d = exp()d volM , C2(M). We call such apair (M,) an weighted Riemannian manifold. We define the weightedLaplacian (also called Witten Laplacian) by

:= ,where is the usual positive Laplacian onM . IfM is closed, then thespectrum of the weighted Laplacian is discrete, where is consid-ered as a self-adjoint operator on L2(M,). We denote its eigenvalues

Supported by a Grant-in-Aid for Scientific Research from the Japan Society for thePromotion of Science.Department of Mathematics, Faculty of Science,Kyoto University, Kyoto 606-8502, JAPANe-mail : [email protected].

1

2 KEI FUNANO

with multiplicity by

0 = 0(M,) < 1(M,) 2(M,) k(M,) .In this paper we study the following problem:

Problem 1.1. How do 1(M,), 2(M,), , k(M,), lie on thereal line?

The above problem amounts to finding the relation among eigenval-ues of the weighted Laplacian.In order to tackle Problem 1.1 let us focus on diameter estimates in

terms of eigenvalues of the weighted Laplacian due to Li and Yau [LY80,Theorem 10] and Cheng [Che75, Corollary 2.2] (see also [Set98]). Com-bining their results one could obtain that k(M,) c(k, n)1(M,)for any natural number k and any closed weighted Riemannian mani-fold (M,) of nonnegative Bakry-Emery Ricci curvature, here c(k, n) isa constant depending only on k and the dimension n ofM . The depen-dence of the constant c(k, n) on n comes from Chengs result. In orderto bypass the dimension dependence of the inequality by Cheng, weconsider the observable diameter ObsDiam((M,);), > 0, intro-duced by Gromov in [Gro99]. The observable diameter comes from thestudy of concentration of measure phenomenon and it might be inter-preted as a substitute of the usual diameter. See Definition 5.5. Theobservable diameter is closely related with the first nontrivial eigen-value of the weighted Laplacian as was firstly observed by Gromov andV. Milman in [GM83]:

ObsDiamR((M,);) 61(M,)

log1

.(1.1)

Under assuming the nonnegativity of Bakry-Emery Ricci curvature,E. Milman obtained the opposite inequality ([Mil10, Mil11, Mil12]):

ObsDiamR((M,);) 1 221(M,)

.

See (2.3), Proposition 2.12, and Lemma 5.6 for the proof of the abovetwo inequalities. Observe that these two inequalities are independent ofthe dimension. One might regard the Gromov-V. Milman inequality asa dimension-free Chengs inequality for k = 1 and also the E. Milmaninequality as a dimension-free Li-Yaus inequality.One of the main results in this paper is the following:

Theorem 1.2. There exists a universal numeric constant c > 0 suchthat if (M,) is a closed weighted Riemannian manifold of nonnegative

EIGENVALUES AND ISOPERIMETRIC CONSTANTS 3

Bakry-Emery Ricci curvature and k is a natural number, then we have

k(M,) exp(ck)1(M,).Theorem 1.2 also holds for a convex domain with C2 boundary in

a closed weighted Riemannian manifold of nonnegative Bakry-EmeryRicci curvature and with the Neumann boundary condition, the proofof which is identical.The crucial point of Theorem 1.2 is that the constant exp(ck) is

independent of the dimension and quantitative. In [FS13, Theorem 1.1]the author proved with Shioya that the fraction k(M,)/1(M,) isbounded from above by some universal constant depending only on k.However the estimate was not quantitative since the proof in [FS13]relies on some compactness argument.In Theorem 1.2, the nonnegativity of Bakry-Emery Ricci curvature is

necessary as was remarked in [FS13]. In fact for any > 0 there existsa closed Riemannian manifold M of Ricci curvature such that2(M)/1(M) 1/ . Taking an appropriate scaling, some dumbbellspace becomes such an example, see [FS13, Example 4.9] for details.The following corollary corresponds to Chengs inequality for general

k:

Corollary 1.3. There exists a universal numeric constant c > 0 suchthat if (M,) is a closed weighted Riemannian manifold of nonnegative


ObsDiamR((M,);) exp(ck)k(M,)

log1

.

Corollary 1.3 follows from Theorem 1.2 together with the Gromov-V. Milman inequality (1.1).In order to treat the k-th eigenvalue we will work on the notion of

separation, which is regarded as a generalization of the concentrationof measure phenomenon (see Subsection 2.2). It tells the informationwhether there exists a pair or not which are not separated in somesense among any k + 1-tuple subsets with a fixed volume. Accord-ing to the work of Chung, Grigoryan, and Yau [CGY96, CGY97], itis related with the information of general eigenvalues of the weightedLaplacian (see Theorem 2.10). Under assuming the nonnegativity of

Bakry-Emery Ricci curvature, we prove that if a nonseparated pair al-ways exists among any k+1-tuple of subsets, then it also holds amongany k-tuple of subsets. In order to prove it, we use the curvature-dimension condition CD(0,) in the sense of Lott-Villani [LV09] andSturm [Stu06a, Stu06], which is equivalent to the nonnegativity of

4 KEI FUNANO

Bakry-Emery Ricci curvature. Idea of the proof of Theorem 1.2 will bediscussed in Section 3 in more detail.

1.2. Multi-way isoperimetric constants. Let (M,) be a closedweighted Riemannian manifold. Recall that Minkowskis (exterior)boundary measure of a Borel subset A ofM , which we denote by +(A),is defined as

+(A) := lim infr0

(Or(A)) (A)r

,

where Or(A) denotes the open r-neighborhood of A. We consider thefollowing geometric quantity:

Definition 1.4 (Multi-way isoperimetric constants). For a naturalnumber k, we define the k-way isoperimetric constant as

hk(M,) := infA0,A1, ,Ak

max0ik

+(Ai)

(Ai),

where the infimum runs over all collections of k+1 non-empty, disjointBorel subsets A0, A1, , Ak of M . h1(M,) is also called the Cheegerconstant.

Note that hk(M,) hk+1(M,) by the definition. We are inter-ested in the distribution of h1(M,), h2(M,), , hk(M,), on thereal line and the relation between k(M,) and hk(M,).The Cheeger-Mazja inequality ([Maz60, Maz61, Maz62] and [Che70],

see [Mil10, Theorem 1.1]) states that

h1(M,) 21(M,).(1.2)

In [LGT12], resolving a conjecture by Miclo [Mic08] (see also [DJM12]),Lee, Gharan, and Trevisan obtained a higher order Cheeger-Mazjainequality for general graphs. Although they proved it for graphs,by an appropriate modification of their proof (e.g., by replacing sumswith integrals), it is also valid for weighted Riemannian manifolds. InAppendix, we will discuss a point that we have to be care when wetreat their argument for the smooth setting.

Theorem 1.5 (Lee et al. [LGT12, Theorem 3.8]). There exists auniversal numerical constant c > 0 such that for all closed weightedRiemannian manifold (M,) and a natural number k we have

hk(M,) ck3k(M,).

Lee et al. proved the above order k3 can be improved to k2 forgraphs ([LGT12, Theorem 1.1]). Since it is uncertain that Lemma 4.7in [LGT12] holds or does not hold for the case of weighted Riemannian


manifolds, the author does not know k3 order in Theorem 1.5 can beimproved to k2 order.The opposite inequality of the Cheeger-Mazja inequality (1.2) was

shown by Buser [Bus82] and Ledoux [Led04]. They proved the existenceof universal numeric constant c > 0 such that

c1(M,) h1(M,)(1.3)

for any closed weighted Riemannian manifold (M,) of nonnegative

Bakry-Emery Ricci curvature. Combining Theorems 1.2 and 1.5 with(1.3) we obtain

hk(M,) . k3k(M,) . k

3 exp(ck)1(M,) . k

3 exp(ck)h1(M,),

where A . B denotes A CB for some universal numeric constantC > 0. Consequently we have the following:

Theorem 1.6. There exists a universal numeric constant c > 0 suchthat if (M,) is a closed weighted Riemannian manifold of nonnegative


hk(M,) k3 exp(ck)h1(M,).In [Mim13] Mimura obtained similar universal inequalities among

multi-way isoperimetric constants for Cayley graphs.Following and extending the heat semigroup argument by Ledoux

[Led04] and E. Milman [Mil12], we obtain the extension of the Buser-Ledoux Theorem:

Theorem 1.7. Assume that a closed weighted Riemannian manifold(M,) has nonnegative Bakry-Emery Ricci curvature. Then for anynatural number k we have

(80k3)1k(M,) hk(M,).

As a consequence hk(M,) andk(M,) are equivalent up to poly-

nomials of k under assuming the nonnegativity of Bakry-Emery Riccicurvature.

1.3. Application to the stability of eigenvalues of the weightedLaplacian and multi-way isoperimetric constants. In [Mil12,Section 5], E. Milman obtained several stability results of the Cheegerconstants on convex bodies. We apply Theorems 1.2 and 1.6 to one ofhis theorem in [Mil12].For a domain with C2 boundary in a complete Riemannian man-

ifold, we denote by k() the k-th eigenvalue of Laplacian with Neu-mann condition.

6 KEI FUNANO

Corollary 1.8. Let K, L be two bounded convex domains in Rn andassume that both K and L have C2 boundary. If

vol(K L) vK vol(K) and vol(K L) vL vol(L),then

k(K) exp(ck)v4K

{log(1 + 1/vL)}2k(L),

and

hk(K) exp(ck)v2K

k3 log(1 + 1/vL)hk(L).

where c > 0 is a universal numeric constant.

In particular, if vol(K) vol(L) vol(K L) then k(K) k k(L)and hk(K) k hk(L). Here A B (resp., A k B) stands for A andB are equivalent up to universal numeric constants (resp., constantsdepending only on k).E. Milman obtained the above corollary for k = 1 ([Mil12, Theo-

rem 1.7]). The above corollary follows from his theorem together withTheorems 1.2 and 1.6.In the same spirit we investigate the (rough) stability property of

eigenvalues of the weighted Laplacian and multi-way isoperimetric con-stants with respect to perturbation of spaces (Section 5). We dis-cuss the case where two weighted manifolds M and N of nonnegativeBakry-Emery Ricci curvature are close with respect to the concentra-tion topology introduced by Gromov in [Gro99]. Roughly speaking,the two spaces M and N are close with respect to the concentrationtopology if 1-Lipschitz functions on M are close to those on N in somesense.

1.4. Organization of the paper. Section 2 collects some back groundmaterial. In Section 3, after explaining some basics of the theory ofoptimal transportation, we prove Theorem 1.2. In Section 4, we proveTheorem 1.7. In Section 5 we study the (rough) stability propertyof eigenvalues of the weighted Laplacian and multi-way isoperimetricconstants with respect to the concentration topology. In Section 6 wediscuss several questions concerning this paper and some conjectureraised in [FS13].

2. Preliminaries

We review some basics needed in the proof of the main theorems.


2.1. Concentration of measure. In this subsection we explain theknown relation among the 1st eigenvalue of the weighted Laplacian,the Cheeger constant, and the concentration of measure in the sense ofLevy and V. Milman ([Lev51], [Mil71]).Let X be an mm-space, i.e., a complete separable metric space with

a Borel probability measure X .

Definition 2.1 (Concentration function, [AM80]). For r > 0 we definethe real number X(r) as the supremum of X(X \ Or(A)), where Aruns over all Borel subsets of X such that X(A) 1/2. The functionX : ( 0,+ ) R is called the concentration function.Lemma 2.2 ([AM80], [Led01, Lemma 1.1]). If X(A) > 0, then

X(X \Or+r0(A)) X(r)for any r, r0 > 0 such that X(r0) < .

The following Gromov and V. Milmans theorem asserts that Poincareinequalities imply appropriate exponential concentration inequalities([GM83], [Led01, Theorem 3.1]).

Theorem 2.3 ([GM83]). Let (M,) be a closed weighted Riemannianmanifold. Then we have

(M,)(r) exp(1(M,)r/3)

for any r > 0. In particular, we have

(M,)(r) exp(h1(M,)r/6)(2.1)for any r > 0.

The second statement (2.1) follows from the first statement togetherwith the Cheeger-Mazja inequality (1.2).

Remark 2.4. Integrating Cheegers linear isoperimetric inequality alsoimplies the second inequality (2.1) (see [MS08, Proposition 1.7]).

In the series of works [Mil10, Mil11, Mil12], E. Milman obtained theconverse of Theorem 2.3 under assuming the nonnegativity of Bakry-Emery Ricci curvature. He proved that a uniform tail-decay of the con-centration function implies the linear isoperimetric inequality (Cheegers

isoperimetric inequality) under assuming nonnegativity of Bakry-EmeryRicci curvature. E. Milmans theorem plays a key role in the proof ofthe main theorems.For an weighted Riemannian manifold (M,), we define the (infinite-

dimensional) Bakry-Emery Ricci curvature tensor as

Ric := RicM +Hess.

8 KEI FUNANO

Theorem 2.5 (E. Milman, [Mil11, Theorem 2.1]). Let (M,) be a

closed weighted Riemannian manifold of nonnegative Bakry-Emery Riccicurvature. If (M,)(r) for some r > 0 and ( 0, 1/2 ), then

h1(M,) 1 2r

.

In particular, we have

1(M,) (1 2

2r

)2.

The key ingredient of E. Milmans approach to the above result is theconcavity of isoperimetric profile under the assumption of the nonnega-tivity of Bakry-Emery Ricci curvature, the fact based on the regularitytheory of isoperimetric minimizers (see [Mil10, Appendix]). See also[Led01] for the heat semigroup approach to Theorem 2.5.

2.2. Separation distance. We define the separation distance whichplays an important role when treating eigenvalues of the weightedLaplacian. The separation distance was introduced by Gromov in[Gro99].

Definition 2.6 (Separation distance). For any 0, 1, , k 0 withk 1, we define the (k-)separation distance Sep(X ; 0, 1, , k) ofX as the supremum of mini 6=j dX(Ai, Aj), where A0, A1, , Ak are anyBorel subsets of X satisfying that X(Ai) i for all i = 0, 1, , k.It is immediate from the definition that if i i for each i =

0, 1, , k, thenSep(X ; 0, 1, , k) Sep(X ; 0, 1, , k).

Note that if the support of X is connected, then

Sep(X ; 0, 1, , k) = 0for any 0, 1, , k > 0 such that

ki=0 i > 1.

For a Borel subset A of an mm-space X we put

A :=X |AX(A)

Lemma 2.7. If A satisfies X(A) , thenSep((A, A); 0, 1, , k) Sep(X ; 0, 1, , k)

for any 0, 1, , k > 0.Proof. Take k+1 Borel subsets A0, A1, , Ak of A such that A(Ai) i for any i. The lemma immediately follows from that X(Ai) X(A)i i.


We denote the closed r-neighborhood of a subset A in a metric spaceby Cr(A).

Lemma 2.8. Let X be an mm-space and put r := Sep(X, 0, 1, , k).Assume that k Borel subsets A0, A1, , Ak1 of X satisfy X(Ai) ifor every i = 0, 1, , k 1 and dX(Ai, Aj) > r for every i 6= j. Thenwe have

X

( k1i=0

Cr(Ai)) 1 k.

Proof. Suppose that for some 0 > 0,

X

( k1i=0

Cr+0(Ai)) 1 k.

Putting Ak := X\k1

i=0 Cr+0(Ai) we have X(Ak) k and dX(Ak, Ai) r + 0 for any i = 0, 1, , k 1. Thus we get

r < mini 6=j

dX(Ai, Aj) Sep(X ; 0, 1, , k) = r,

which is a contradiction. Hence X(k1

i=0 Cr+(Ai)) > 1 k for any > 0. Letting 0 we obtain the conclusion. The following lemma asserts that exponential concentration inequal-

ities and logarithmic 2-separation inequalities are equivalent:

Lemma 2.9. Let X be an mm-space.

(1) If X satisfies

Sep(X ; , ) 1Clog

c

(2.2)

for any > 0, then we have X(r) c exp(Cr) for any r > 0.(2) Conversely, if X satisfies X(r) c exp(C r) for any r > 0,

then we have

Sep(X ; , ) 2C

logc

for any > 0.

Proof. (1) Assume that X satisfies (2.2) and let A X be a Borelsubset such that X(A) 1/2. For r > 0 we put := X(X \Or(A)).Since

r dX(X \Or(A), A) Sep(X ; , 1/2) Sep(X ; , ) 1Clog

c

,

we have c log(Cr), which gives the conclusion of (1).

10 KEI FUNANO

(2) Assuming that X(r) c exp(C r), we take two Borel sub-sets A,B X such that X(A) , X(B) , and dX(A,B) =Sep(X ; , ). Let r be any positive number satisfying

X(r) c exp(C r) < ,i.e.,

r >1

C log

c

.

Since X(A) , by Lemma 2.2, we have1 X(O2r(A)) X(r) < .

Hence we have

X(O2r(A) B) > (1 ) + 1 = 0,which yields Sep(X ; , ) = dX(A,B) 2r. Letting r C 1 log(c/)we obtain (2).

Theorem 2.3 together with Lemma 2.9 (2) implies that for any closedweighted Riemannian manifold (M,) we have

Sep((M,); , ) 61(M,)

log1

.(2.3)

Chung, Grigoryan, and Yau generalized the above inequality in thefollowing form:

Theorem 2.10 (Chung et al. [CGY97, Theorem 3.1]). Let (M,) bea closed weighted Riemannian manifold. Then, for any k N and any0, 1, , k > 0, we have

Sep((M,); 0, 1, , k) 1k(M,)

maxi 6=j

log( eij

).

Combining Theorems 1.5 and 2.10 we obtain the following proposi-tion:

Proposition 2.11. There exists a universal numeric constant c > 0such that for all closed weighted Riemannian manifold (M,), a naturalnumber k, and 0, 1, , k > 0, we have

Sep((M,); 0, 1, , k) ck3

hk(M,)maxi 6=j

loge

ij.

We end this subsection reformulating Theorem 2.5 in terms of theseparation distance for later use.


Proposition 2.12. Let (M,) be a closed weighted Riemannian mani-

fold of nonnegative Bakry-Emery Ricci curvature. Then, for any > 0,we have

Sep((M,); , ) 1 2h1(M,)

and

Sep((M,); , ) 1 221(M,)

.

Proof. We prove only the first assertion. The proof of the secondassertion is identical to the first one. According to Theorem 2.5, ifr ( 0, ) and ( 0, 1/2 ) satisfy

r . There exists A M such that (A) 1/2and (M \Or(A)) > . Hence we haver = dM(A,M \Or(A)) Sep((M,); 1/2, ) Sep((M,); , ).

Combining the above inequality with (2.4) gives the conclusion.

Since Sep((M,); , ) diamM , the first inequality of Theorem2.12 recovers the Li-Yau inequality [LY80].

2.3. Three distances between probability measures. Let X be acomplete separable metric space. We denote by P(X) the set of Borelprobability measures on X .

Definition 2.13 (Prohorov distance). Given two measures , P(X) and 0, we define the Prohorov distance di(, ) as theinfimum of > 0 such that

(C(A)) (A) and (C(A)) (A) (2.5)for any Borel subsets A X .For any 0, the function di is a complete separable distance

function on P(X). If > 0, then the topology on P(X) determinedby the Prohorov distance function di coincides with that of the weakconvergence (see [Bil99, Section 6]). The distance functions di for all > 0 are equivalent to each other. Also it is known that if (C(A)) (A) for any Borel subsets A of X , then di(, ) . In otherwords, the second inequality in (2.5) follows from the first one (see[Bil99, Section 6]).For (x, y) X X , we put proj1(x, y) := x and proj2(x, y) :=

y. For two finite Borel measures and on X , we write if

12 KEI FUNANO

(A) (A) for any Borel subset A X . A finite Borel measure pi onX X is called a partial transportation from P(X) to P(X)if (proj1)(pi) and (proj2)(pi) . Note that we do not assumepi to be a probability measure. For a partial transportation pi from to , we define its deficiency def pi by def pi := 1 pi(X X). Given > 0, the partial transportation pi is called an -transportation from to if it is supported in the subset

{(x, y) X X | dX(x, y) }.Definition 2.14 (Transportation distance). Let 0. For two prob-ability measures , P(X), we define the transportation distanceTra(, ) between and as the infimum of > 0 such that thereexists an -transportation pi from to satisfying def pi .The following theorem is due to V. Strassen.

Theorem 2.15 ([Vil03, Corollary 1.28], [Gro99, Section 312.10]). For

any > 0, we have

Tra = di .

Let (X, dX) be a complete metric space. We indicate by P2(X) theset of all Borel probability measures P(X) such that

XdX(x, y)

2d(y) < +

for some x X .Definition 2.16 ((L2-)Wasserstein distance). For two probability mea-sures , P2(X), we define the L2-Wasserstein distance dW2 (, )between and as the infimum of

(XX

dX(x, y)2dpi(x, y)

)1/2,

where pi P2(XX) runs over all couplings of and , i.e., probabilitymeasures pi with the property that pi(AX) = (A) and pi(X A) =(A) for any Borel subset A X . It is known that this infimum isachieved by some transport plan, which we call an optimal transportplan for dW2 (, ).

If the underlying space X is compact, then the topology on P(X)induced from the L2-Wasserstein distance function coincides with thatof the weak convergence (see [Vil03, Theorem 7.12]).


3. Proof of Theorem 1.2

In order to prove Theorem 1.2 we need to explain some useful toolsfrom the theory of optimal transportation. Refer to [Vil03, Vil08] formore details.Let (X, dX) be a metric space. A rectifiable curve : [0, 1] X is

called a geodesic if its arclength coincides with the distance dX((0), (1))and it has a constant speed, i.e., parameterized proportionally to the ar-clength. We say that a metric space is a geodesic space if any two pointsare joined by a geodesic between them. It is known that (P2(X), dW2 )is compact geodesic space as soon as X is ([Stu06, Proposition 2.10]).LetM be a close Riemannian manifold. For two probability measures

0, 1 P2(M) which are absolutely continuous with respect to d volM ,there is a unique geodesic (t)t[0,1] between them with respect to theL2-Wasserstein distance function dW2 ([McC01, Theorem 9]).For an mm-space X let us denote by the set of minimal geodesics

: [0, 1] X endowed with the distanced(1, 2) := sup

t[0,1]dX(1(t), 2(t)).

Define the evaluation map et : X for t [0, 1] as et() := (t).A probability measure P() is called a dynamical optimal trans-ference plan if the curve t := (et), t [0, 1], is a minimal geodesicin (P2(X), dW2 ). Then pi := (e0 e1) is an optimal coupling of 0and 1, where e0 e1 : X X is the endpoints map, i.e.,(e0 e1)() := (e0(), e1()).Lemma 3.1 ([LV09, Proposition 2.10]). If (X, dX) is locally compact,then any minimal geodesic (t)t[0,1] in (P2(X), dW2 ) is associated witha dynamical optimal transference plan , i.e., t = (et).

Let and be two probability measures on a set X . We definethe relative entropy Ent() of with respect to as follows. If isabsolutely continuous with respect to , writing d = d, then

Ent() :=

M

log d,

otherwise Ent() :=.Definition 3.2 (Curvature-dimension condition, [LV09], [Stu06a, Stu06]).Let K be a real number. We say that an mm-space satisfies thecurvature-dimension condition CD(K,) if for any 0, 1 P2(X)there exists a minimal geodesic (t)t[0,1] in (P2(X), dW2 ) from 0 to 1

14 KEI FUNANO

such that

EntX (t) (1 t) EntX (0) + tEntX (1)K

2(1 t)t dW2 (0, 1)2

for any t [0, 1].In the above definition, assume that both 0 and 1 are absolutely

continuous with respect to X . Then Jensens inequality applied to theconvex function r 7 r log r gives

log X(Supp t)

(3.1)

(1 t)M

0 log 0dX tM

1 log 1dX +Kt(1 t)

2dW2 (0, 1)

2,

where 0 and 1 are densities of 0 and 1 with respect to X respec-tively. In particular, for two Borel subsets A,B X with X(A), X(B) >0, we have

log X(Supp t)

(3.2)

(1 t) log X(A) + t logX(B) + Kt(1 t)2

dW2

( X |AX(A)

,X |BX(B)

)2

([Stu06],[Oht13]).

Theorem 3.3 ([CMS01, CMS06], [vRS05], [Stu05]). For a completeweighted Riemannian manifold (M,), we have Ric K for someK R if and only if (M,) satisfies CD(K,).Theorem 1.2 follows from the following key theorem together with

Theorem 2.10 and Proposition 2.12.

Theorem 3.4. Let (M,) be a closed weighted Riemannian manifold

of nonnegative Bakry-Emery Ricci curvature. If (M,) satisfies

Sep((M,); , , , k+1 times

) 1D

log1

2(3.3)

for any > 0, then we have

Sep((M,); , , , k times

) cD

log1

2(3.4)

for any > 0 and for some universal numeric constant c > 0.


The idea of the proof of Theorem 3.4 is the following. It turnsout that it is enough to prove (3.4) for sufficiently small > 0 andsufficiently large c > 0. We suppose the converse of this, i.e.,


) >c

Dlog

1

2

for sufficiently small > 0 and sufficiently large c > 0. Put :=(c/D) log(1/). By the definition of the separation distance there existsk Borel subsets A0, A1. , Ak1 M such that mini 6=j d(Ai, Aj) > and (Ai) for any i. If we choose the constant c large enough sothat


, 100) Sep((M,); 100, 100, , 100 k+1 times

) /100,

then by Lemma 2.8 we have

( k1

i=0

C/100(Ai)) 1 100.

It means that if > 0 is sufficiently small, the measure of the setk1i=0 C/100(Ai) is nearly 1. Although it is not true, we assume that

( k1

i=0

C/100(Ai))= 1(3.5)

in order to tell the idea of the proof. Putting A := C/100(A0) and B :=k1i=1 C/100(Ai), we have M = AB, AB = , (A) , (B) ,

and d(A,B) /2.Let (t)t[0,1] be a geodesic from A := (1/(A))|A to with respect

to dW2 . For sufficiently small t > 0 we have d(x,A) < /2 d(A,B) forany x Supp t, which gives Suppt A. This leads a contradictionsince by (3.2) we have

log(A) log(Suppt) (1 t) log(A) + log (M),which implies log (A) 0. Although (3.5) is always not true, we showbelow that the above idea can be accomplished by controlling separatedsubsets and estimating average distances between them.

Proof of Theorem 3.4. It suffices to prove that there exist two universalnumeric constants c0, 0 > 0 such that


) c0D

log1

2(3.6)

16 KEI FUNANO

for any 0. In fact, if 1/2, then the left-hand side of the aboveinequality is zero and there is nothing to prove. In the case where0 < 1/2, by (3.6) we have


) Sep((M,); 0, 0, , 0 k times

)

c0 log

120

D log 12

log1

2

c0 log

120

D log 4log

1

2,

which implies the conclusion of the theorem.Suppose the contrary to (3.6), i.e.,


) >c1D

log1

2,(3.7)

where c1 > 0 is a sufficiently large universal numeric constant and > 0 is a sufficiently small number. Both the largeness of c1 andthe smallness of will be specified later. Note that the assumption(3.7) immediately gives k < 1 (otherwise, the left-hand side of (3.7)is zero). We denote the right-hand side of (3.7) by , i.e.,

:=c1D

log1

2.

Claim 3.5. If c1 > 0 (resp., > 0) in (3.7) is large enough (resp.,small enough), then there exist two closed subsets B0, B1 M suchthat B0 B1, /4 (B0) 1/2, (B1) 1 6, and

dM(B0, B1 \B0) c2max{,

1(M,)

}

for some universal numeric constant c2 > 0.

Proof. The assumption (3.7) implies the existence of k Borel subsetsA0, A1, , Ak1 M such that (Ai) for any i and

dM(Ai, Aj) for any i 6= j.If < 1/8 and c1 8, then by (3.3) we have


, 1/8) Sep((M,); , , , k+1 times

) 1D

log1

2

8.


Hence Lemma 2.8 yields

( k1

i=0

C/8(Ai)) 7

8.(3.8)

Note that

dM(C/8(Ai), C/8(Aj)) /4(3.9)for any i 6= j. According to Proposition 2.12 we take X0, X1 M suchthat

(Xi) 12

4(i = 0, 1)(3.10)

and

dM(X0, X1) 81(M,)

.(3.11)

Set Y := X0 X1. By (3.10) we have (Y ) 1 2 and thus(Y C/8(Ai)) (Y Ai)

(1

2

)+ 1

2

for each i = 0, 1, , k 1. Suppose that (Xi C/8(Al)) < /4 forsome i {0, 1} and for any l = 0, 1, , k 1. Then we have

(Xi

k1l=0

C/8(Al)) k

4 0, we get

Sep((M,);

2,

2, ,

2 k times

, 6)

16

(

4

)

provided that c1 in (3.7) is large enough. Put B0 := C/16(A0) and

B1 := C/16(A0)C/16(A1). We may assume that (C/16(A0)) 1/2.

Thanks to Lemma 2.8 it is easy to check that (B1) 1 6. By(3.12) and (3.13), we see that B0 and B1 possess the other desiredproperties.

We consider two Borel probability measures Bi, i = 0, 1, defined by

Bi :=|Bi(Bi)

.

The following claim is essentially due to Gromov [Gro99] (see also[FS13, Claim 5.10]). He used it in the context of the convergence theoryof mm-spaces without detailed proof. Since our context is different fromhis one, we include the proof for the concreteness of this paper. Theproof below is shorter than the one in [FS13, Claim 5.10].

Claim 3.6 ([Gro99, Section 312.47]). There exist a universal numeric

constant c4 > 0 and a coupling pi of B0 and B1 such that

pi({

(x, y) M M | dM(x, y) >c4 log

12

1(M,)

}) 6.

Proof. We use the identity di(B0 , B1) = Tra(B0 , B1) (Theorem2.15). Put := c4

1(M,)log 1

2, where c4 > 0 is a numeric universal

constant which will be determined later. We shall prove that

B1(C(A)) B0(A) 6(3.14)for any Borel subset A B1, which implies the claim. In fact, applying(3.14) to Theorem 2.15 gives that there exists a -transportation pi0from B0 to B1 such that def pi0 6. If def pi0 = 0, then we setpi := pi0. If def pi0 > 0, then set

pi := pi0 +1

def pi0(B0 (proj1)pi0) (B1 (proj2)pi0).

It is easy to check that pi fulfills the desired property.


To prove (3.14) we may assume that B0(A) 6, which yields thatB1(A) (A) 6(B0) 7/4.

Using Lemma 2.7 and (2.3) we choose c4 > 0 so that

Sep((B1, B1);

7

4,7

4

) Sep

((M,); (1 6)

7

4, (1 6)

7

4

)

c41(M,)

log1

2(= ).

Lemma 2.8 implies that

B1(C(A)) 17

4 1 6 B0(A) 6,

which is (3.14).

We set

:={(x, y) M M | d(x, y) c4 log

12

1(M,)

}.

We consider two Borel probability measures 0 := a(proj1)(pi|) and1 := a(proj2)(pi|), where a := pi()1. By Claim 3.6 we have

1 a 11 6(3.15)

and

dW2 (0, 1)

2 aMM

d(x, y)2dpi|(x, y)

{ c4 log 121(M,)

}2.(3.16)

Take an optimal dynamical transference plan such that (ei) = ifor each i = 0, 1. Putting r := dM(B0, B1 \B0), we consider

t := { Supp | dM(e0(), et()) r/2}.By (3.16) we have

r2

4( \ t) dW2 ((e0), (et))2 = t2 dW2 (0, 1)2

{ c4t log 121(M,)

}2.

According to Claim 3.5 we thus get

(t) 1c5t

2(log 1

2

)22

(3.17)

for some universal numeric constant c5 > 0. For s [0, 1] we puts := (es)

|t(t)

. By the definition of s we obtain the following.

Claim 3.7. Supp t B1 B0.

20 KEI FUNANO

By using Claim 3.7, we get

log (B0) +6

(B0) log(B0) + log

(1 +

6

(B0)

)(3.18)

= log((B0) + 6)

log{(Supp t B1) + (Supp t \B1)}= log(Supp t)

Note that (s)s[0,1] is a geodesic between 0 and 1. Since

i =(ei)|t(t)

(ei)(t)

=i

(t) a

(t)(proji+1)pi =

a

(t)Bi

(3.19)

for i = 0, 1, each i is absolutely continuous with respect to , andespecially the above geodesic (s)s[0,1] is unique. For each i = 0, 1, wewrite di = id. By (3.1), we get

log (Supp t) (1 t)M

0 log 0d tM

1 log 1d.(3.20)

For a subset A M we denote by 1A the characteristic function of A,i.e., 1A(x) := 1 if x A and 1A(x) := 0 if x M \ A.Claim 3.8. We have

i log i ct1Bi(Bi)

logct1Bi(Bi)

(i = 0, 1),

where ct := a/(t).

Proof. By (3.19) we have i (ct/(Bi))1Bi. Since ct 1 and u log u v log v for any two positive numbers u, v such that u v and v 1,we obtain the claim.

Combining Claim 3.8 with (3.18) and (3.20) we have

log (B0) +6

(B0)

(1 t)M

ct1B0(B0)

logct1B0(B0)

d tM

ct1B1(B1)

logct1B1(B1)

d

= ct log ct + ct(1 t) log(B0) + ctt log (B1).


Substituting t := 3, we thereby obtain

log(1/2) + 42(3.21)

log (B0) + 6

3(B0)

ct3

log ct +ct 13

(1 3) log(B0) + ct log(B1).

Using (3.15) and (3.17) we estimate each term on the right-side of theabove inequalities as

ct log ct3

=a

(t) log a log (t)

3

11 6

(1

c56(log 1

2

)22

)1

13

(log

1

1 6 log(1

c56(log 1

2

)22

))

11 6

(1 c54

(log

1

2

)2)1 2(3 + c5

(log

1

2

)2),

ct 13

log (B0) a (t)

3(t)log

2

1

14 1 + c54

(log 1

2

)23(t)

log2

1 + c5(1 4)

(log 1

2

)2(1 4)

(1 c54

(log 1

2

)2) log 2,

and

|ct log (B1)| a(t)

log1

1 6 26

(1 6)(1 c54

(log 1

2

)) .These estimates imply the right-side of the inequalities (3.21) is closeto zero for sufficiently small > 0. Since the left-side of the inequality(3.21) is about log(1/2) < 0 for sufficiently small > 0, this is acontradiction. This completes the proof of the theorem.

22 KEI FUNANO

4. Proof of Theorem 1.7

On a closed weighted Riemannian manifold (M,), denote by (Pt)t0the semigroup associated with the infinitesimal generator . For eacht 0, Pt : C(M) C(M) is a bounded linear operator and weextend the action of Pt to L

p() (p 1).The following gradient estimate of the heat semigroup is due to Bakry

and Ledoux [BL96]. One might regard it as a dimension-free Li-Yauparabolic gradient inequality [LY86].

Lemma 4.1 (Bakry-Ledoux, [BL96, Lemma 4.2]). Let (M,) be a

closed weighted Riemannian manifold of Bakry-Emery Ricci curvaturebounded from below by a nonpositive real number K. Then for anyt 0 and f C(M) we have

c(t)|Pt(f)|2 Pt(f 2) (Pt(f))2,where

c(t) :=1 exp(2Kt)

K (= 2t if K = 0).

Corollary 4.2. If (M,) has nonnegative Bakry-Emery Ricci curva-ture, then for any t 0, p 2, and f C(M), we have

|Pt(f)|Lp() 12tfLp().

From Corollary 4.2 Ledoux obtained the following lemma:

Lemma 4.3 (Ledoux, [Led04, (5.5)]). Assume that (M,) has non-

negative Bakry-Emery Ricci curvature. Then for any f C(M), wehave

f Pt(f)L1() 2t|f |L1().

Proof of Theorem 1.7. Take any k+1 non-empty, disjoint Borel subsetsA0, A1, , Ak M . We may assume that (A0) (A1) (Ak), and thus

k1i=0

(Ai) 1 1k + 1

and (Ai) 1/2 for any i = 0, 1, , k 1.

We put t := 4k(k + 1)/k(M,). We shall prove that there exists i0,0 i0 k 1, such that

+(Ai0) (80k3)1k(M,)(Ai0).(4.1)


For each i = 0, 1, , k1, let 1Ai,(x) := min{0, 1 1 d(x,Ai)} denotea Lipschitz approximation of 1Ai. Note that

(C(Ai)) (Ai)

M

|1Ai,|d,

where for a Lipschitz function f :M R and x M , we put

|f |(x) := lim supyx

|f(y) f(x)|dM(y, x)

.

Letting 0, by Lemma 4.3 we have2t+(Ai) 1Ai Pt(1Ai)L1().

Since the right-side of the above inequality can be written as

Ai

(1 Pt(1Ai))d+M\Ai

Pt(1Ai)d

= 2((Ai)

Ai

Pt(1Ai)d)

= 2((Ai)(1 (Ai))

M

(Pt(1Ai) (Ai))(1Ai (Ai))d),

we obtain

2t+(Ai)(4.2)

2((Ai)(1 (Ai))

M

(Pt(1Ai) (Ai))(1Ai (Ai))d).

Observe that Pt(1Ai) (Ai), i = 0, 1, , k 1, are linearly in-dependent and orthogonal to constant functions on M . Thus theRayleigh quotient representation of k(M,) yields that there exista0, a1, , ak1 R such that

k(M,) |(k1i=0 ai(Pt(1Ai) (Ai)))|2L2()k1i=0 ai(Pt(1Ai) (Ai))2L2() .(4.3)

Put f0 :=k1

i=0 ai1Ai . We consider the following two cases: (I) f0 Mf0dL2() 2f0 Pt(f0)L2(), (II) f0

Mf0dL2() 2f0

Pt(f0)L2().

24 KEI FUNANO

We prove that the case (I) cannot happen from the our choice of t.Suppose that (I) holds. In this case we get

k1i=0

ai(Pt(1Ai) (Ai))L2() =Pt(f0)

M

f0dL2()

(4.4)

12

f0 M

f0dL2()

.

We estimate the right-side of the above inequality from below:

Claim 4.4. We haveM

( k1i=0

ai(1Ai (Ai)))2d 1

k + 1

k1i=0

a2i

M

(1Ai (Ai))2d.

Proof. Since M

(1Ai (Ai))2d = (Ai)(1 (Ai))(4.5)and

M

( k1i=0

ai(1Ai (Ai)))2d =

k1i=0

a2i(Ai)( k1

i=0

ai(Ai))2,

it suffices to prove

(1 1

k + 1

) k1i=0

a2i(Ai) +1

k + 1

k1i=0

a2i(Ai)2

( k1i=0

ai(Ai))2.

(4.6)

Since ( k1i=0

ai(Ai))2

=( k1

j=0

(Aj))2( k1

i=0

(Ai)k1j=0 (Aj)

ai

)2

( k1

j=0

(Aj))2 k1

i=0

(Ai)k1j=0 (Aj)

a2i

(1 1

k + 1

) k1i=0

a2i(Ai),

we have (4.6). This completes the proof of the claim.

Claim 4.4 together with (4.3) and (4.4) implies the existence of i0,0 i0 k 1, such that

k(M,)1Ai0 (Ai0)2L2() 4k(k + 1)|Pt(1Ai0 )|2L2().


Using Corollary 4.2 and t = 4k(k + 1)/k(M,) we obtain

k(M,)1Ai0(Ai0 )2L2() 2k(k + 1)

t1Ai0 (Ai0)2L2()

= 21k(M,)1Ai0 (Ai0)2L2(),which is a contradiction.Since (II) holds, Lemma 4.3 yields

1

4

f0 M

f0d2L2()

Pt(f0) f02L2()(4.7)

kk1i=0

a2i Pt(1Ai) 1Ai2L2()

kk1i=0

a2i Pt(1Ai) 1AiL1()

k2t

k1i=0

a2i |1Ai|L1()

= k2t

k1i=0

a2i+(Ai).

According to Claim 4.4 and (4.7), there exists i0, 0 i0 k 1, suchthat

1Ai0 (Ai0)2L2() 4k(k + 1)2t+(Ai0).

Thus we getM

(Pt(1Ai0 ) (Ai0))(1Ai0 (Ai0))d 1Ai0 (Ai0)2L2() 4k(k + 1)

2t+(Ai0).

Since (Ai0) 1/2, it follows from (4.2) that(8k2 + 8k + 1)

2t+(Ai0) 2(Ai0)(1 (Ai0)) (Ai0).

Recalling that t = 4k(k + 1)/k(M,), we finally obtain

+(Ai0) k(M,)

(16k(k + 1) + 2)2k(k + 1)

(Ai0) k(M,)

80k3(Ai0),

which implies (4.1). This completes the proof of the theorem.

26 KEI FUNANO

Remark 4.5. From the proof of [BL96] Bakry-Ledouxs lemma (Lemma

4.1) follows from the following Bakry-Emery type L2-gradient estimate:

|Pt(f)|2(x) e2KtPt(|f |2)(x)(4.8)for any Lipschitz function f and any x X . Gigli, Kuwada, andOhta proved the gradient estimate (4.8) for compact finite-dimensionalAlexandrov spaces satisfying CD(K,) ([GKO13, Theorem 4.3]). HereAlexandrov spaces are metric spaces whose sectional curvature isbounded from below in the sense of the triangle comparison property.In particular the same argument in this section implies that Theorem1.7 holds for compact finite-dimensional Alexandrov spaces satisfyingCD(0,). Refer to [KMS01] for the Laplacian on Alexandrov spaces.We remark that Theorem 1.5 holds for compact finite-dimensionalAlexandrov spaces from the proof of [LGT12]. Consequently the k-theigenvalue of Laplacian and the k-way isoperimetric constant are equiv-alent up to polynomials of k for compact finite-dimensional Alexandrovspaces satisfying CD(0,). In particular it is also valid for compactfinite-dimensional Alexandrov spaces of nonnegative curvature, sincesuch spaces satisfy CD(0,) ([Pet11], [ZZ10]).

5. Rough stability of eigenvalues of the weightedLaplacian and multi-way isoperimetric constants

We first review the concentration topology. Recall that the Hausdorffdistance between two closed subsets A and B in a metric space X isdefined by

dH(A,B) := inf{ > 0 | A C(B), B C(A) }.Let (I, ) be a probability space. We denote by F(I,R) the space of

all -measurable functions on I. Given 0 and f, g F(I,R), weput

me(f, g) := inf{ > 0 | (|f g| > ) },where (|f g| > ) := ({x I | |f(x) g(x)| > }). Note that, ifany two functions f, g F(I,R) with f = g a.e. are identified to eachother, then me is a distance function on F(I,R) for any 0 and itstopology on F(I,R) coincides with the topology of the convergence inmeasure for any > 0. The distance functions me for all > 0 aremutually equivalent.Let d be a semi-distance function on I, i.e., a nonnegative symmetric

function on I I satisfying the triangle inequality. We indicate byLip1(d) the space of all 1-Lipschitz functions on I with respect to d .


Note that Lip1(d) is a closed subset in (F(I,R),me) for any 0.For 0 and two semi-distance functions d and d on I, we define

HL1(d , d ) := dH(Lip1(d),Lip1(d )),where dH is the Hausdorff distance function in (F(X,R),me). HL1is a distance function on the space of all semi-distance functions onX for all 0, and the two distance functions HL1 and HL1are equivalent to each other for any , > 0. We denote by L theLebesgue measure on R.For any mm-spaceX there exists a Borel measurable map : [ 0, 1 )

X with L = X (see [Kec95, Theorem 17.41]). We call such a map a parameter of X . Note that a parameter of X is not unique in general.For a parameter ofX , we define a function dX : [ 0, 1 )[ 0, 1 ) Rby dX(s, t) := dX((s), (t)) for any s, t [ 0, 1 ).Definition 5.1 (Observable distance function). For two mm-spaces Xand Y we define

HL1(X, Y ) := infHL1(X dX , Y dY ),where the infimum is taken over all parameters X : [ 0, 1 ) X andY : [ 0, 1 ) Y .We say that two mm-spaces are isomorphic to each other if there is

a measure preserving isometry between the spaces. Denote by X thespace of isomorphic classes of mm-spaces. The function HL1 is adistance function on X for any 0. Note that HL1 and HL1are equivalent to each other for any , > 0.

Definition 5.2 (Concentration topology). We say that a sequence ofmm-spaces Xn, n = 1, 2, , concentrates to an mm-space Y if Xnconverges to Y as n with respect to H1L1. The topology onthe set X induced by the observable distance function is called theconcentration topology.

The term concentration topology comes from the following: We saythat a sequence of mm-spaces {Xn} is a Levy family if limn Xn(r) =0 for any r > 0. Due to Levys lemma ([Lev51], [Led01, Proposition1.3]) we obtain the following:

Proposition 5.3 ([Gro99]). A sequence {Xn}n=1 of mm-spaces is aLevy family if and only if it concentrates to the one-point mm-space.

For example, the sequence of n-dimensional unit spheres in Rn+1, n =1, 2, , concentrates to the one-point space by Levys result([Lev51]).The concentration topology is strictly weaker than the measured

Gromov-Hausdorff topology on the space of mm-spaces ([Fun08]). We

28 KEI FUNANO

mention that the concentration topology coincides with the measuredGromov-Hausdorff topology on the set of mm-spaces satisfying CD(K,N)for fixed K and N < +. In fact, the set becomes compact with re-spect to the measured Gromov-Hausdorff topology because we havethe doubling condition with a uniform doubling constant under thecondition CD(K,N).Answering a conjecture by Fukaya in [Fuk87], Cheeger and Colding

proved the continuity of eigenvalues of Laplacian on Riemmanian man-ifolds with respect to the measured Gromov-Hausdorff topology underthe condition CD(K,N) for fixed K,N R ([CC00]). We consideran analogy of the above Cheeger-Colding result with respect to theconcentration topology:

Corollary 5.4. There exists a universal numeric constant c > 0 satis-fying the following. Let {(Mn, n)} be a sequence of closed weightedRiemannian manifolds of nonnegative Bakry-Emery Ricci curvatureand assume that the sequence concentrates to a closed weighted Rie-mannian manifold (M, ). Then for any natural number k we have

lim supn

max{ k(Mn, n)k(M, )

,k(M, )

k(Mn, n)

} exp(ck)(5.1)

and

lim supn

max{ hk(Mn, n)hk(M, )

,hk(M, )

hk(Mn, n)

} k3 exp(ck).(5.2)

Note that dimension of Mn may diverge to infinity as n.The rest of this subsection is devoted to prove Corollary 5.4. For the

proof we first recall the definition of observable diameter introduced byGromov in [Gro99]:

Definition 5.5 (Observable diameter). Let > 0. We define thepartial diameter

diam(X , 1 )of X as the infimum of diamA over all Borel subsets A X with(A) 1 . Define the observable diameter

ObsDiamR(X ;)ofX as the supremum of diam(fX , 1) over all 1-Lipschitz functionsf : X R.The idea of the observable diameter comes from the quantum and

statistical mechanics, i.e., we think of X as a state on a configurationspace X and f is interpreted as an observable.


The next lemma expresses the relation between the observable di-ameter and the separation distance. The proof of the lemma is foundin [Fun06, Subsection 2.2]

Lemma 5.6 ([Gro99]). Let X be an mm-space. For any , > 0 with > , we have

(1) Sep(X ; , ) ObsDiamR(X ;),(2) ObsDiamR(X ;2) Sep(X ; , ).

Lemma 5.7. Let X, Y be two mm-spaces and assume that H1L1(X, Y ) < < 1. Then for any ( , 1 ) we have

ObsDiamR(Y ;) ObsDiamR(X ;( )) + 2 .Proof. The condition H1L1(X, Y ) < implies the existence of twoparameters X : [ 0, 1 ) X and Y : [ 0, 1 ) Y such that

dH(X Lip1(X), Y Lip1(Y )) < .

Hence, for any f Lip1(Y ), there exists g Lip1(X) such thatL(|f Y g X | > ) < .

Take a Borel subset A R such that gX(A) 1 + anddiam(gX , 1 ( )) = diamA. Putting

B := f Y ({|f Y g X | } (g Y )1(A)),we find

fY (B) (1 ) + (1 + ) 1 = 1 .Given s, t {|f Y g X | } (g Y )1(A) we have

|f Y (s) f Y (t)| |f Y (s) g X(s)|+ |g X(s) g X(t)|

+ |g X(t) f Y (t)| diamA+ 2 ,

which implies diam(fY , 1 ) diamA + 2 . This completes theproof.

Lemma 5.8. Let (M,M) and (N, N) be two closed weighted Rie-

mannian manifolds of nonnegative Bakry-Emery Ricci curvature suchthat H1L1((M,M), (N, N)) < 1/2. Assume that two positive num-bers , satisfies H1L1((M,M), (N, N)) < < 1/2 and + < 1/2.Then we have

1(N, N) 1(M,M){ 2 1(M,M) 6 log(14 2 2)

}2(5.3)

30 KEI FUNANO

and

h1(N, N) h1(M,M) h1(M,M) 6 log(14 2 2)

.(5.4)

Proof. Combining (2.3), Lemmas 5.6 and 5.7 gives that for any > we have

ObsDiamR((N, N);) ObsDiamR((M,M);( )) + 2 Sep

((M,M);

2

, 2

)+ 2

61(M,M)

log2

+ 2 .

Lemma 5.6 again yields

Sep((N, N); , ) 61(M,M)

log2

+ 2 .

As in the proof of Lemma 2.9 (1) we obtain

(N,N )(r) +2 exp(611(M,M)(r 2 ))

for any r > 2 . By subsutituting

r := 2 6 log(14

2

2)

1(M,M)

we obtain (N,N )(r) 21 . Applying Theorem 2.5 then impliesthe inequality (5.3). The proof of (5.4) is similar and we omit it.

Proof of Corollary 5.4. Due to Theorem 2.5 we have supnN 1(Mn, n) 0 such thati (k + 1) i for any i, we have

Sep(Y ; 0, 1, , k) Sep(X ; 0, 1, , k) + 2 .


Proof. Take k+1 Borel subsets A0, A1, , Ak Y such that Y (Ai) i for any i and mini 6=j dY (Ai, Aj) = Sep(Y ; 0, 1, , k). SinceH1L1(X, Y ) < there exist two parameters X : [ 0, 1 ) X andY : [ 0, 1 ) Y such that H1L1(X dX , Y dY ) < . For eachi = 0, 1, , k, we put fi(x) := dY (x,Ai). Since each fi is 1-Lipschitz,the condition H1L1(X dX , Y dY ) < implies the existence of k + 11-Lipschitz functions gi : X R, i = 0, 1, , k, such that me1(fi Y , gi X) < . Putting

I :=

ki=0

{|fi Y gi X | }

we have L(I) 1(k+1) . For each i = 0, 1, , k we define Bi Xas Bi := X(

1Y (Ai) I). Note that X(Bi) L(1Y (Ai) I)

i (k+1) . For any ai 1Y (Ai) I , aj 1Y (Aj) I, i 6= j, we getdX(X(ai), X(aj)) |gi(X(ai)) gj(X(aj))|

|fi(Y (ai)) fj(Y (aj))| 2 dY (Ai, Aj) 2 ,

which implies that

mini 6=j

dX(Bi, Bj) mini 6=j

dY (Ai, Aj) 2 = Sep(Y ; 0, 1, , k) 2 .This completes the proof.

Corollary 5.10. Assume that a sequence {Xn} of mm-spaces concen-trate to an mm-space Y . Then we have

lim infn

Sep(Xn; 0,

1, , k) Sep(Y ; 0, 1, , k)(5.5)

and

lim supn

Sep(Xn; 0, 1, , k) Sep(Y ; 0, 1, , k)(5.6)

for any 0, 1, , k, 0, 1, , k > 0 such that i > i.6. Questions

In this section we raise several questions which are concerned withthis paper. We also discuss conjecture which was posed in [FS13].Throughout this section, unless otherwise stated, we will always assumethat (M,) is a closed weighted Riemannian manifold of nonnegative

Bakry-Emery Ricci curvature.

Question 6.1. Independent of k, is it possible to bound k+1(M,)/k(M,)or hk+1(M,)/hk(M,) from above by a universal numeric constant ?

32 KEI FUNANO

Masato Mimura asked me about the fraction of k+1(M,)/k(M,).Theorem 1.2 leads to the above question for eigenvalues of the weightedLaplacian. Due to Theorems 2.10 and 3.4, in order to give an affirma-tive answer to Question 6.1 for eigenvalues it suffices to extend E. Mil-mans theorem (Theorem 2.5) in terms of k(M,) and the k-separationdistance, i.e., any k-separation inequalities imply appropriate lowerbounds of the k-th eigenvalue k(M,). Or more weakly, it suffices toprove that any logarithmic k-separation inequalities of the form (3.3)give appropriate estimates of the k-th eigenvalue k(M,) from below.This can also be considered as an extension of [GRS11, Theorem 1.14].In [GRS11] Gozlan, Roberto, and Samson proved that any exponen-tial concentration inequalities imply appropriate Poincare inequalitiesunder assuming CD(0,). Notice that by Lemma 2.9 exponentialconcentration inequalities are nothing but logarithmic 2-separation in-equalities.For multi-way isoperimetric constants, we also need to improve k3

order in Proposition 2.11 to some universal numeric constant. Thefollowing integration argument makes possible to improve k3 order butit is not logarithmic separation inequalities:

Proposition 6.2. Let (M,) be a closed weighted Riemannian mani-fold and k a natural number. Then for any > 0 we have

Sep((M,); , , , k+1 times

) 2log 2

log(2/)hk(M,)

.

Proof. Let A0, A1, , Ak be k+1 Borel subsets ofM such that (Ai) for any 0 i k. Our goal is to prove the following inequality:

D := mini 6=j

dM(Ai, Aj) 2log 2

log(2/)hk(M,)

.(6.1)

In order to prove (6.1) we may assume that each Ai is given by afinite union of open balls. For r [0, D/2) we put Bi := Or(Ai),i = 0, 1, , k 1, and Bk := M \

k1i=0 Bi. By the definition of

hk(M,), we have +(Bi0) hk(M,)(Bi0) for some i0. Assume first

that i0 = k. Since each Bi consists of a finite union of open balls weobtain

k1i=0

+(Bi) = +(Bk) hk(M,)(Bk) hk(M,).


In the case where i0 k 1, we getk1i=0

+(Bi) +(Bi0) hk(M,)(Bi0) hk(M,)

Combining the above two inequalities implies that

( k1

i=0

Or(Ai))

( k1i=0

Ai

)=

r0

k1i=0

+(Os(Ai))ds hk(M,)r,

which yields

(M \

k1i=0

Or(Ai)) (1 hk(M,)r)

(M \

k1i=0

Ai

).(6.2)

What follows is a straightforward adaption of Gromov-V. Milmansargument in [GM83, Theorem 4.1]. Put := (2hk(M,))

1. If r,then there exists a natural number j such that j r < (j + 1) .Iterating (6.2) k times shows

(M \Or

( k1i=0

Ai

))

(M \Oj

( k1i=0

Ai

))

(1 hk(M,) )(M \O(j1)

( k1i=0

Ai

))

(1 hk(M,) )j(M \

k1i=0

Ai

)

(1 hk(M,) )j= exp(j log 2) exp((r/ ) log 2)= exp(hk(M,)r2 log 2).

If r < , then we have

(M \Or

( k1i=0

Ai

)) 1 2 2 1 r 2 exp(hk(M,)r2 log 2)

Put r := D/2. Combining the above two inequalities we obtain

(Ak) (M \OD

2

( k1i=0

Ai

)) 2 exp(hk(M,)D log 2),

34 KEI FUNANO

which implies (6.1). This completes the proof.

Question 6.3. What is the right order ofk(M,)/hk(M,), k(M,)/1(M,),

and hk(M,)/h1(M,) in k? Especially can we bound k(M,)/1(M,)and hk(M,)/h1(M,) from above by some polynomial function of k ?

The following two questions are concerned with the stability of eigen-values of the weighted Laplacian and multi-way isoperimetric con-stants.

Question 6.4. Is it true that if two convex domains K,L Rn satisfyvol(K) vol(L), then k(K) k(L) or hk(K) hk(L)?Question 6.5. Can we get the stability of eigenvalues of the weightedLaplacian and multi-way isoperimetric constants with respect to theconcentration topology ? Or more weakly can we replace exp(ck) andk3 exp(ck) in Corollary 5.4 with some universal numeric constant ?

In view of Corollary 5.10 an extension of E. Milmans theorem for thek-separation distance and the k-th eigenvalue would imply the latterquestion in Question 6.5.In [FS13, Conjecture 6.11] we raised the following conjecture.

Conjecture 6.6. For any natural number k there exists a positiveconstant Ck depending only on k such that if X is a compact finite-dimensional Alexandrov space of nonnegative curvature, then we have

k(X) Ck1(X).Since Theorems 1.5 and 1.7 hold for compact finite-dimensional Alexan-

drov spaces of nonnegative curvature, the above question amounts tosaying the existence of Ck such that hk(X) Ckh1(X).We remark that Theorem 2.10 holds for compact finite-dimensional

Alexandrov spaces. In fact, the only we need in the proof is the Davies-Gaffney heat kernel estimate

A

B

pt(x, y)d(x)d(y) (A)(B) exp

( d

2(A,B)

4t

)

for any Borel subsets A,B and asymptotic expansion of heat kernel byeigenvalues and eigenfunctions of Laplacian ([CGY96]). These are truefor compact finite-dimensional Alexandrov spaces ([Stu95], [KMS01]).However it is not known the corresponding theorem of E. Milmanstheorem (Theorem 2.5) for Alexandrov spaces. Note that we usedTheorem 2.5 in the proof of Theorem 3.4. In order to give an affirmativeanswer to Conjecture 6.6, it suffices to prove that any concentrationinequalities imply appropriate exponential concentration inequalities


under assuming CD(0,) or Theorem 3.4 holds for general CD(0,)spaces by Gozlan-Roberto-Samsons theorem [GRS11, Theorem 1.14].

Acknowledgments.The author would like to thank to Professors Alexan-der Bendikov, Alexander Grigoryan, Emanuel Milman, Kazuhiro Kuwae,Karl Theoder Sturm, and Nathael Gozlan for their comments and theirinterests of this paper. He also thanks to Professor Masato Mimura forseveral discussion. A part of this work was done while the authorvisited Bonn university and Bielefeld university.

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7. Appendix

The only point we need to be care when we prove Lee-Gharan-Trevisans theorem (Theorem 1.5) for the smooth setting is the fol-lowing lemma:

Lemma 7.1 ([LGT12, Lemma 2.1]). Let X be an mm-space and f :X Rn a Lipschitz map. Then there exists a closed subset A of Xsuch that A Supp f and

+X(A)

X(A) 2|f |L2(X)fL2(X)

.

Proof. For any positive real number t we put

At := {x X | |f(x)|2 t}.Note that At Supp f for any t > 0 and

0

X(At)dt = f2L2(X )(7.1)The co-area inequality ([BH97, Lemma 3.2]) implies that

0

+X(At)dt M

|(|f |2)|(x)dX(x)(7.2)

2M

|f(x)||f |(x)dX(x) 2fL2(X )|f |L2(X ).

Combining (7.1) with (7.2) gives0+X(At)dt

0X(At)dt

2|f |L2(X )fL2(X ),

which implies the conclusion of the lemma.

1. Introduction1.1. Eigenvalues of the weighted Laplacian1.2. Multi-way isoperimetric constants1.3. Application to the stability of eigenvalues of the weighted Laplacian and multi-way isoperimetric constants1.4. Organization of the paper

2. Preliminaries2.1. Concentration of measure2.2. Separation distance2.3. Three distances between probability measures

3. Proof of Theorem ??4. Proof of Theorem ??5. Rough stability of eigenvalues of the weighted Laplacian and multi-way isoperimetric constants6. QuestionsReferences7. Appendix

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