AN OPTIMAL DIMENSION-FREE UPPER BOUND FOR EIGENVALUE ...

arXiv:1405.2213v2 [math.DG] 21 May 2014

AN OPTIMAL DIMENSION-FREE UPPER BOUND FOR EIGENVALUE RATIOS SHIPING LIU Abstract. On a closed weighted Riemannian manifold with nonnega´ tive Bakry-Emery Ricci curvature, it is shown that the ratio of the k-th to first eigenvalues of the weighted Laplacian is dominated by 641k2 , using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of k here is optimal, and hence answers an open question of Funano. This approach works still on a compact finite-dimensional Alexandrov space of nonnegative curvature and proves affirmatively a conjecture of Funano and Shioya asserting a dimension free upper bound for eigenvalue ratios in that setting.

1. Introduction Let (M, µ) be a closed weighted Riemannian manifold, where µ is a Borel probability measure of the form dµ = e−V dvolM , V ∈ C 2 (M ) and volM stands for the Riemannian volume measure of M . In this note, we study the eigenvalues of the corresponding weighted Laplacian ∆µ , which are listed with multiplicity as below, 0 = λ0 (M, µ) < λ1 (M, µ) ≤ λ2 (M, µ) ≤ · · · ≤ λk (M, µ) ≤ · · · ր ∞. We prove the following theorem. Theorem 1.1. For any closed weighted Riemannian manifold (M, µ) of ´ nonnegative Bakry-Emery Ricci curvature and any natural number k, we have   16e 2 2 k λ1 (M, µ). (1) λk (M, µ) ≤ e−1 For (M, µ) as stated in the above theorem, it is known from the work of Cheng [10] and Li-Yau [23] (see also Setti [29]) on estimating eigenvalues by diameter that there exists a numeric constant C such that λk (M, µ) ≤ Cn2 k2 λ1 (M, µ),

(2)

where n is the dimension of M . The first dimension-free estimate of these eigenvalue ratios was discovered by Funano-Shioya [13] through investigating spectral characterizations of L´evy families. They proved that there exists a constant Ck depending only on k such that λk (M, µ) ≤ Ck λ1 (M, µ), 1

(3)

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and propose a conjecture asserting that (3) still holds for compact finitedimensional Alexandrov space of nonnegative curvature (see Conjecture 6.11 in [13] and also Conjecture 6.6 in [12]). Later, Funano [12] proved a quantitative version of (3), showing that one can find a numeric constant c > 0 such that Ck in (3) can be taken as Ck = eck . Funano further asked for the right order of λk (M, µ)/λ1 (M, µ) in k, or especially whether this ratio can be dominated by a polynomial function of k or not (see Question 6.3 in [12]). Theorem 1.1 answers Funano’s question. In fact, the order of k in (1) is optimal. This can be seen from the following example (see e.g. Section 4.E.2 in [14]). Example 1.2. The eigenvalues of the circle TL1 of length L are   2 2 4π k : k = 0, 1, 2, . . . , L2

where each non-zero eigenvalue has multiplicity 2.

It is worth mentioning here Weyl’s asymptotic formula (see e.g. p.155 in [7], [16]) to see the optimality for the order of k. It reads 2  n k as k → ∞, (4) λk (M, µ) ∼ c(n) µ(M )

where c(n) is a constant depending on dimension n. Note further that while the Cn2 in (2) explodes as the dimension n increases, the constant in our estimate (1) is smaller than 641. We also mention Funano and Shioya’s observation that the nonnegativity of Ricci curvature is necessary (See Example 4.9 in [13]). Moreover, our approach is steadily extendable to a more general setting. In fact, we have the following result. Theorem 1.3. For any compact finite-dimensional Alexandrov space X of nonnegative curvature and any natural number k, we have   16e 2 2 k λ1 (X). (5) λk (X) ≤ e−1 This verifies a strong version of the Conjecture 6.11 in Funano-Shioya [13].

1.1. An approach via Cheeger constant. We prove Theorem 1.1 by connecting λ1 (M, µ) with λk (M, µ) via Cheeger constant h1 (M, µ). The first ingredient is due to a recent progress of Kwok-Lau-Lee-GharanTrevisan [20] in theoretical analysis for spectral clustering algorithms. In order to justify the stability of this popular heuristic algorithm rigorously, they proved a so-called improved Cheeger inequality for finite graphs. They

AN UPPER BOUND FOR EIGENVALUE RATIOS

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also indicated a corresponding inequality for a closed Riemannian manifold. But their constant depends on the dimension of the manifold. In fact, an appropriate extension of their proof to the setting of weighted Riemannian manifolds gives the following dimension-free estimate. Theorem 1.4. On a closed weighted Riemannian manifold (M, µ), we have for any natural number k, √ λ1 (M, µ) . h1 (M, µ) ≤ 8 2k p λk (M, µ)

(6)

Note that when k = 1, (6) reduces to Cheeger inequality [8] with a larger constant. The proof of improved Cheeger inequality for graphs depends on the fact that an eigenvalue of the normalized graph Laplacian is no greater than 2. This is fortunately not continuous space case. Comparing √ √ needed for√ with the graph case h1 ≤ 10 2(k + 1)λ1 / λk in [20], the constant 8 2 here is smaller. This is mainly due to a special argument for continuous space (see Claim 3.1). Then we observe that one can combine Theorem 1.4 with the dimensionfree Buser inequality [6] due to Ledoux [21] to prove Theorem 1.1. Theorem 1.5. On a closed weighted Riemannian manifold (M, µ) of non´ negative Bakry-Emery Ricci curvature, we have e − 1p λ1 (M, µ). h1 (M, µ) ≥ √ (7) 2e

We comment that the constant here is larger than 1/6 used in [21](see the formula (5.8) there). This is because in [21] the two cases that lower Ricci curvature bound being zero/a negative number are handled simultaneously. Once we restrict ourselves to nonnegative curvature case, the constant can be as in (7). Now (6) and (7) implies Theorem 1.1. For recent developments in spectral theory of Markov operators on probability spaces thanks to another significant progress in analyzing spectral clustering of Lee-Gharan-Trevisan [22], see [27, 31, 24].

1.2. Applications. Theorme 1.1 has important applications. We use it to improve the higher-order Buser-Ledoux inequality and the higher-order Gromov-Milman inequality established by Funano [12] (see Theorem 4.1, Theorem 4.5 below). The latter inequality relates eigenvalues with the observable diameter introduced by Gromov [17]. Actually, the improved version of it obtained here can implies Cheng’s diameter estimates [10] up to a universal constant. Theorem 1.1 also implies that the ratio of the k-th to first isoperimetric constants (see Definition 2.1) can be bounded by a polynomial function of k, answering part of Question 6.3 of [12]. This approach also has interesting applications in spectral graph theory, for which we will discuss in a subsequent note.

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2. Preliminaries Let (M, µ) be a closed weighted Riemannian manifold. The weighted Laplacian ∆µ is given by ∆µ := ∆ − ∇V · ∇, where ∆ is the Laplace-Beltrami operator on M . For its basic spectral theory, see e.g. Section 2 of [29]. For any functions f ∈ W 1,2 (µ), we denote its Rayleigh quotient by ´ |∇f (x)|2 dµ(x) ´ . R(f ) := M 2 M f (x) dµ(x)

In the following, we only need deal with Lipschitz functions as they are dense in W 1,2 (µ). ´ The Bakry-Emery Ricci curvature tensor (see e.g. [1]) is defined as Ricµ := Ric + HessV,

where Ric is the usual Ricci curvature tensor on M . For any Borel subset A ⊆ M , its boundary measure µ+ (A) is defined as µ+ (A) := lim inf r→0

µ(Or (A)) − µ(A) , r

where Or (A) := {x ∈ M : d(x, a) < r for some a ∈ A} is the open rneighborhood of A. Define φ(A) :=

µ+ (A) . µ(A)

For a nonnegative Lipschitz function f : M → R, denote Mf (t) := {x ∈ M : f (x) > t}. Then let φ(f ) := inf t∈R≥0 φ(Mf (t)). Definition 2.1 (Multi-way isoperimetric constants [26, 12]). For a natural number k, the k-th isoperimetric constant is defined as hk (M, µ) :=

inf

max φ(Ai ),

A0 ,A1 ,...,Ak 0≤i≤k

(8)

where the infimum is taken over all collections of k + 1 non-empty, disjoint Borel subsets A0 , A1 , . . . , Ak of M . Note that hk (M, µ) ≤ hk+1 (M, µ) and h1 (M, µ) is the Cheeger constant. Lemma 2.2. There exist two nonnegative disjointly supported Lipschitz functions f0 and f1 such that R(f0 ) = R(f1 ) = λ1 (M, µ). Proof. Let f be the corresponding eigenfunction of λ1 (M, µ), and f0 (x) := max{f (x), 0}, f1 (x) := max{−f (x), 0}.

(9)

AN UPPER BOUND FOR EIGENVALUE RATIOS

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´ Since M f (x)dµ(x) = 0, we know both f0 and f1 are non-trivial. By definition, we have for i = 0, 1, ˆ ˆ ˆ fi2 dµ fi f dµ = λ1 (M, µ) h∇fi , ∇f idµ = λ1 (M, µ) M M M ˆ h∇fi , ∇fi idµ. = M

That is, R(fi ) = λ1 (M, µ).



A strong extension of this simple fact is the following lemma due to LeeGharan-Trevisan [22], Funano [12], Miclo [27]. Lemma 2.3. For any natural number k, there exist k + 1 nonnegative disjointly supported Lipschitz functions f0 , f1 , . . . , fk and a numeric constant C such that for each 0 ≤ i ≤ k, R(fi ) ≤ Ck6 λk (M, µ).

(10)

The following lemma is a direct corollary of the co-area inequality (Lemma 3.2 in Bobkov-Houdr´e [3], see also Funano [12]). Lemma 2.4. For any nonnegative Lipschitz function f , we have ´ |∇f (x)|dµ(x) ´ φ(f ) ≤ M . M f (x)dµ(x) Proof. It is directly to observe ˆ ˆ f (x)dµ(x) = M

∞

µ(Mf (t))dt.

(12)

|∇h(x)|dµ(x).

(13)

0

The co-area inequality tells ˆ ˆ ∞ + µ (Mf (t))dt ≤ 0

(11)

M

Combining (12) and (13) gives ´∞ + ´ µ (Mf (t))dt |∇f (x)|dµ(x) 0 ´ φ(f ) ≤ ´ ∞ . ≤ M M f (x)dµ(x) 0 µ(Mf (t))dt



3. Proof of Theorem 1.4 In this section, we present details of a direct extension of Kwok-Lau-LeeGharan-Trevisan [20] to prove Theorem 1.4. For convenience, we decide to write down a self-contained proof here. Particular efforts are made to improve the involving constants. Theorem 3.1. For any nonnegative Lipschitz function f : M → R, we have √ R(f ) . (14) φ(f ) ≤ 8 2k p λk (M, µ)

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This theorem will follow immediately from the two lemmas below. For any given values t0 , t1 , . . . , tl ∈ R, we define a function ψt0 ,t1 ,...,tl : R → R such that for any x ∈ R, ψt0 ,t1 ,...,tl (x) = arg

min

ti ∈{t0 ,t1 ,...,tl }

|x − ti |.

Given a nonnegative Lipschitz function f , and a sequence of real values 0 = t0 ≤ t1 ≤ · · · ≤ t2k = T := maxx∈M f (x), we then have a (2k + 1)-step function gk defined as gk (x) := ψt0 ,t1 ,...,t2k (f (x)).

(15)

Such step function approximations of f share the following property. Lemma 3.2. For any nonnegative Lipschitz function f , and a sequence of real values 0 = t0 ≤ t1 ≤ · · · ≤ t2k = T , we have φ(f ) ≤ 8k

p

R(f )

k f − gk kL2 (µ) . k f kL2 (µ)

(16)

Proof. We define a function η : R → R as η(x) := |x − ψt0 ,t1 ,...,t2k (x)|, ∀x ∈ R, and then a funtion h : M → R as ˆ f (x) η(t)dt, ∀x ∈ M. h(x) := 0

Observe that for any two points x, y ∈ M , h(x) ≥ h(y) if and only if f (x) ≥ f (y). Hence we obtain φ(h) = φ(f ). Recall by Lemma 2.4, we have ´ |∇h(x)|dµ(x) ´ φ(h) ≤ M . (17) M h(x)dµ(x) For the numerator, we estimate ˆ ˆ |∇h(x)|dµ(x) =

! ˆ f (x) η(t)dt dµ(x) ∇ M 0

M

=

ˆ

M

|∇f (x)||η(f (x))|dµ(x) ≤k |∇f | kL2 (µ) k f − gk kL2 (µ) .

(18)

Observe that for any x ∈ M , we can find an integer i such that ti < f (x) ≤ ti+1 and hence ˆ f (x) i−1 ˆ tj+1 X η(t)dt + η(t)dt h(x) = j=0

≥

tj

ti

i−1 X (tj+1 − tj )2 j=0

4

+

(f (x) − ti )2 . 4

(19)

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Meanwhile, we have by Cauchy-Schwarz inequality,  2 i−1 X f 2 (x) =  (tj+1 − tj ) + (f (x) − ti ) j=0

≤ 2k

i−1 X (tj+1 − tj )2 + 2k(f (x) − ti )2 .

(20)

j=0

Combining (19) and (20) implies

1 2 f (x), ∀x ∈ M. 8k Now (18) and (21) together implies (16). h(x) ≥

(21) 

In fact, one can find nice step function approximations of f . Lemma 3.3. For any nonnegative Lipschitz function f : M → R, there exists a (2k + 1)-step approximation gk such that k f − gk k2L2 (µ) ≤

2R(f ) k f k2L2 (µ) . λk (M, µ)

(22)

Proof. Let us construct t0 , t1 , . . . , t2k inductively. First, set t0 = 0. Suppose we have already set the values of t0 , t1 , . . . , ti−1 , then we find ti in the following way. If there exists t ≥ ti−1 such that ˆ R(f ) k f k2L2 (µ) , (23) |f (x) − ψti−1 ,t (f (x))|2 dµ(x) = C0 := kλk (M, µ) {x:ti−1 k, there exists a numeric constant C such that λk (M, µ) . (27) hk (M, µ) ≤ Clk6 p λl (M, µ) 4. Applications and extensions

4.1. Some direct applications. Theorem 1.1 has important applications. Combining it with Buser-Ledoux inequality (7), we obtain the following improvement of Funano’s higher-order Buser-Ledoux inequality (Theorem 1.7 in [12]). Theorem 4.1. On a closed Riemannian manifold (M, µ) of nonnegative ´ Bakry-Emery Ricci curvature, we have for any natural number k (e − 1)2 1 p λk (M, µ). (28) hk (M, µ) ≥ √ 16 2e2 k

We remark that the constant above [12].

(e−1)2 √ 16 2e2 k

≥

1 57k

improves the

1 80k 3

in

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Proof. This follows from e − 1 e − 1p e − 1p λ1 (M, µ) ≥ √ · λk (M, µ), hk (M, µ) ≥ h1 (M, µ) ≥ √ 2e 2e 16ek

where we used monotonicity of hk (M, µ), Theorem 1.5 and Theorem 1.1.  Following Theorem 1.6 in Funano [12], we obtain the multi-way isoperimetric constant ratio estimate below. It improves Funano’s exponential upper bound into a polynomial one. Theorem 4.2. There exists a universal numeric constant C such that if (M, µ) is a closed weighted Riemannian manifold of nonnegative Bakry´ Emery Ricci curvature and k is a natural number, then we have p hk (M, µ) ≤ Ck log(1 + k)h1 (M, µ). (29)

Proof. We need the following shifted higher-order Cheeger inequality (Theorem 4.1 in [22] and Theorem 12 in [27]) saying that p (30) hk (M, µ) ≤ C1 λ2k (M, µ) log(1 + k),

for some universal numeric constant C1 . Combining with Theorem 1.1 and Theorem 1.5, we arrive at (29).  Remark 4.3. The shifted higher-order Cheeger inequality (30) was proved in finite graph setting by Lee-Gharan-Trevisan [22]. It was then extended to Riemannian setting via a probability theoretic approximation procedure of Miclo [27]. However, a direct proof of it for Riemannian setting is still unknown (see [12]). Remark 4.4. In the above two results, estimates of the two ratios, p λk (M, µ)/hk (M, µ) and hk (M, µ)/h1 (M, µ),

are improved. However, the author does not know whether they are in the right order of k or not (see Question 6.3 in Funano [12]). Observe in Example 1.2, we have hk (TL1 ) = 2k/L (by an argument similar to Proposition 7.3 in [24]), that is , the first ratio is of order 0 and the second of order 1 in k. Observable diameter with parameter κ > 0 is an important concept introduced by Gromov [17]. For (M, µ), the partial diameter diam(µ; 1 − κ) is defined as the infimum of the diameter of A over all Borel subsets A ⊆ M with µ(A) ≥ 1 − κ. Then the observable diameter is defined to be ObsDiam((M, µ); −κ) := sup{diam(f∗ µ; 1 − κ)}, where the supremum is taken over all 1-Lipschitz functions f : M → R. It is important in characterizing measure concentration phenomena. Funano [12] obtained the following version of Gormov-Milman [18] inequality. ObsDiam((M, µ); −κ) ≤ p

2 6 log . κ λ1 (M, µ)

(31)

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As observed in [12], (31) together with Theorem 1.1 provides the following result, which can be considered as a dimension-free Cheng’s inequality. Theorem 4.5. For any closed weighted Riemannian manifold (M, µ) of ´ nonnegative Bakry-Emery Ricci curvature and any natural number k, we have 2 152k log . (32) ObsDiam((M, µ); −κ) ≤ p κ λk (M, µ) Note that 152k here improves the eck in Corollary 1.3 of [12]. Recall under the assumption of nonnegative Ricci curvature, Cheng (Corollary 2.2 in [10]) proved for any natural number k, diam(M ) ≤

p

2n(n + 4)k p , λk (M )

(33)

where n is the dimension of M . As kindly pointed to us by Kei Funano, (32) in fact implies (33) up to a universal constant, using the equivalence of observable diameter and separation distance (see e.g. Lemma 5.6 in [12]) and an argument of Chung-Grigor’yan-Yau (see (3.15) in [11]). Observe that one can not expect an exact dimension-free version of (33), thinking of the example of unite spheres {S n }∞ n=1 . We also remark that Theorem 1.1 holds still for a convex domain with C 2 boundary in a closed weighted Riemannian manifold of nonnegative Bakry´ Emery Ricci curvature and with Neumann boundary condition, using an identical proof. Hence those stability results for eigenvalues and multi-way isoperimetric constants by Funano [12] (see Corollary 1.8 and Corollary 5.4 there) can be improved in the way that the exponential functions of k are replaced by polynomial functions of k accordingly (see Question 6.5 in [12]). 4.2. Extensions to Alexandrov spaces. An Alexandrov space (X, d) of curvature bounded from below is a complete geodesic metric space which locally satisfies the Toponogov triangle comparison theorem for sectional curvature. The Hausdorff dimension of an Alexandrov space is an integer or infinity. See [4, 5, 28] for more details about Alexandrov geometry. We consider a compact n-dimensional Alexandrov space X equipped with the n-dimensional Hausdorff measure Hn in this subsection. We refer to Section 2 of [15] for a nice review with detailed references about the Sobolev space W 1,2 (Hn ) on an Alexandrov space. Among them we mention that the set of Lipschitz functions is dense in W 1,2 (Hn ) (Theorem 1.1 in [19]). For a Lipshitz function f : X → R, the weak gradient vector ∇f (x) satisfies p

h∇f (x), ∇f (x)i = lim sup y→x

|f (x) − f (y)| , a.e. x. d(x, y)

(34)

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(See [19] for precise definitions of the weak gradient vector and the inner product h·, ·i.) The Dirichlet energy is defined as ˆ h∇f, ∇gidHn , for f, g ∈ W 1,2 (Hn ). E(f, g) := X

This definition coincides with Cheeger’s energy functional [9] in terms of minimal generalized upper gradient. The spectral theory about Laplacian ∆X associated with (E, W 1,2 (Hn )) on an Alexandrov space was studied in [19]. In particular, ∆X has discrete spectrum consisting of 0 = λ1 (X) < λ2 (X) ≤ · · · ≤ λk (X) ≤ · · · ր ∞ with finite multiplicity. Moreover, the corresponding eigenfunctions are Lipschitz continuous (Theorem 4.4 of [15]). The min-max principle we used in (26) follows from a standard argument. Note that the co-area inequality of Bobkov-Houdr´e [3] holds in a general metric measure space where the measure is not atomic, and the modulus of a function’s gradient there coincide with (34). Bearing in mind those facts mentioned above, we see Theorem 1.4 extends steadily to a compact finite-dimensional Alexandrov space of curvature bounded from below. As pointed out by Funano (Remark 4.5 in [12]), thanks to the Bakry´ Emery type gradient estimate for the heat kernel on an Alexandrov space due to Gigli-Kuwada-Ohta [15], Theorem 1.5 holds on a compact finitedimensional Alexandrov space of nonnegative curvature. This then verifies Theorem 1.3. Finally we remark that Theorem 1.3 is still true on a weighted Alexandrov space under assuming CD(0, ∞) in the sense of Lott-Sturm-Villani ([25, 30]) since Gigli-Kuwada-Ohta gradient estimate extends to that setting [15]. Acknowledgements I am very grateful to Professor Norbert Peyerimhoff for many helpful discussions. I also thank Professors Kei Funano and Feng-Yu Wang for their interests and valuable comments. This work was supported by the EPSRC Grant EP/K016687/1 ”Topology, Geometry and Laplacians of Simplicial Complexes”. References [1] D. Bakry, Functional inequalities for Markov semigroups, Probability measures on groups: recent directions and trends, S. G. Dani and P. Graczyk (Editors), Tata Inst. Fund. Res., Mumbai (2006), pp. 91-147. [2] P. H. B´erard, Spectral geometry: direct and inverse problems, with appendixes by G´erard Besson, and by B´erard and Marcel Berger, Lecture Notes in Mathematics, 1207, Springer-Verlag, Berlin, 1986. xiv+272 pp. [3] S. G. Bobkov and C. Houdr´e, Isoperimetric constants for product probability measures, Ann. Probab. 25 (1997), no. 1, 184-205.

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[24] S. Liu, Multi-way dual Cheeger constants and spectral bounds of graphs, arXiv:1401.3147, Janurary 2014. [25] J. Lott and C. Villani, Ricci curvature for metric measure spaces via optimal transport, Ann. of Math. 169 (2009), 903-991. [26] L. Miclo, On eigenfunctions of Markov processes on trees, Probab. Theory Related Fields 142 (2008), no. 3-4, 561-594. [27] L. Miclo, On hyperboundedness and spectrum of Markov operators, hal-00777146, version 3, December 2013. [28] Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994), no. 3, 629-658. [29] A. G. Setti, Eigenvalue estimates for the weighted Laplacian on a Riemannian manifold, Rend. Sem. Mat. Univ. Padova 100 (1998), 27-55. [30] K.-T. Sturm, On the geometry of metric measure spaces I., Acta Math. 196 (2006) 65-131. [31] F.-Y. Wang, Criteria of spectral gap for Markov operators, J. Funct. Anal. 266 (2014), 2137-2152. Department of Mathematical Sciences, Durham University, DH1 3LE Durham, United Kingdom E-mail address: [email protected]