The Steklov and Laplacian spectra of Riemannian manifolds with boundary

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Abstract

Given two compact Riemannian manifolds M1 and M2 such that their respective boundaries Σ1 and Σ2 admit neighbourhoods Ω1 and Ω2 which are isometric, we prove the existence of a constant C such that |σk(M1)σk(M2)|C for each kN. The constant C depends only on the geometry of Ω1Ω2. This follows from a quantitative relationship between the Steklov eigenvalues σk of a compact Riemannian manifold M and the eigenvalues λk of the Laplacian on its boundary. Our main result states that the difference |σkλk| is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω1Ω2.

Introduction

Let M be a smooth compact Riemannian manifold of dimension n+12, with nonempty boundary Σ. The Steklov eigenvalue problem on M is to find all numbers σR for which there exists a nonzero function uC(M) which satisfies{Δu=0 in M,un=σu on Σ. Here n is the outward unit normal along Σ and Δ=div is the Laplace-Beltrami operator acting on C(M). The Steklov problem has a discrete unbounded spectrum0=σ1σ2σ3+, where each eigenvalue is repeated according to its multiplicity. For background on this problem, see [12], [21] and references therein.

Let H:C(Σ)C(M) be the harmonic extension operator: the function u=Hf satisfies u=f on Σ and Δu=0 in M. The Steklov eigenvalues of M are the eigenvalues of the Dirichlet-to-Neumann (DtN) map D:C(Σ)C(Σ), which is defined byDf=nHf. The DtN map is a first order self-adjoint elliptic pseudodifferential operator [28, pages 37–38]. Its principal symbol is given by p(x,ξ)=|ξ|, while that of the Laplace operator ΔΣ:C(Σ)C(Σ) is |ξ|2. Standard elliptic theory [16], [27] then implies thatσkλk2π(kωnVoln(Σ))1/n as k, where ωn is the volume of the unit ball B(0,1)Rn. It follows that manifolds M1 and M2 which have isometric boundaries satisfy σk(M1)σk(M2) as k.

It was proved in [22] that the full symbol of the DtN map is determined by the Taylor series of the Riemannian metric of M in the normal direction along its boundary Σ (see also [25]). Consider a closed Riemannian manifold (Σ,gΣ) and two compact Riemannian manifolds (M1,g1) and (M2,g2) with the same boundary Σ=M1=M2 such that g1|Σ=g2|Σ=gΣ. If the metrics g1 and g2 have the same Taylor series on Σ, then M1 and M2 have asymptotically equivalent1 Steklov spectra:σk(M1)=σk(M2)+O(k) as k. See [11, Lemma 2.1]. In particular, limkσk(M1,g1)σk(M2,g2)=0.

The asymptotic behaviour described by (1) or (2) does not contain any information regarding an individual eigenvalue σk. In order to obtain such information, stronger hypothesis are needed. Indeed on any smooth compact Riemannian manifold (M,g0) with boundary, there exists a family of Riemannian metrics (gϵ)ϵR+ such that gϵ=g0 on a neighbourhood Ωϵ of Σ=M, while for each kN,limϵσk(M,gϵ)=0. See [2] and [5] for two such constructions. If n2, there also exists a family of Riemannian metrics such that gϵ=g0 on a neighbourhood Ωϵ of Σ andlimϵσ2(M,gϵ)=+. See [2] for the construction of these families. In this last construction, the neighbourhoods Ωϵ are shrinking to Σ as ϵ:ϵ(0,)Ωϵ=Σ. However, if the manifolds M1 and M2 are uniformly isometric near their boundary, the situation is completely different.

Theorem 1

Given two compact Riemannian manifolds with boundary M1 and M2 such that their respective boundaries Σ1 and Σ2 admit neighbourhoods Ω1 and Ω2 which are isometric, there exists a constant C, which depends explicitly on the geometry of Ω1Ω2, such that |σk(M1)σk(M2)|C for each kN.

Not only the difference |σk(M1)σk(M2)| is bounded for each given kN, but the bound is uniform: the constant C does not depend on k. The existence of C could be obtained from a Dirichlet-to-Neumann bracketting argument, but this would not lead to an explicit expression.

Theorem 1 is a manifestation of the principle stating that Steklov eigenvalues are mostly sensitive to the geometry of a manifold near its boundary. The more information we have on the metric g on, and near, the boundary Σ, the more we can say about the Steklov spectrum of M. Knowing the metric on Σ leads to the asymptotic formula (1), then knowing its Taylor series in the normal direction along Σ leads to the refined asymptotic formula (2), and finally the knowledge of the metric on a neighbourhood of the boundary provides the extra information needed to control individual eigenvalues in Theorem 1.

Remark 2

Let be the number of connected components of the boundary Σ. Under the hypothesis of Theorem 1, it is well known that for k the following holds:C1σk(M1)σk(M2)C. This was used for instance in [2], [6]. Theorem 1 is stronger since it implies11+Cσk(M1)σk(M1)σk(M2)1+Cσk(M2). It is also known that σk(Mi)σkNσN, where σkN refer to the mixed Neumann-Steklov problem on Ω1Ω2. See [2] for details.

Theorem 1 follows from our main result, which is a quantitative comparison between the Steklov eigenvalues of M and the eigenvalues of the Laplace operator ΔΣ on its boundary Σ, which are denoted 0=λ1λ2. Let nN and let α,β,κ,κ+R and h>0 be such that αβ and κκ+. Consider the class M=M(n,α,β,κ,κ+,h) of smooth compact Riemannian manifolds M of dimension n+1 with nonempty boundary Σ, which satisfy the following hypotheses:

  • (H1)

    The rolling radius2 of M satisfies h¯:=roll(M)h.

  • (H2)

    The sectional curvature K satisfies αKβ on the tubular neighbourhoodMh¯={xM:d(x,Σ)<h¯}.

  • (H3)

    The principal curvatures of the boundary Σ satisfy κκiκ+.

The main result of this paper is the following.

Theorem 3

There exist explicit constants A=A(n,α,β,κ,κ+,h) and B=B(n,α,κ,h) such that each manifold M in the class M satisfies the following inequalities for each kN,λkσk2+Aσk,σkB+B2+λk. In particular, for each kN, |σkλk|<max{A,2B}.

Under the hypotheses of Theorem 3 one can takeA=nh˜+|α|+κ2 and B=12h¯+n2|α|+κ2, where h˜h¯ is a positive constant depending on h¯, β, and κ+. See (10) for the precise definition. Various other values for the constants A and B will be given under more restrictive assumptions in Section 1.4.

This theorem is in the spirit of the very nice result of [26] where a similar statement is proved for Euclidean domains. The dependence of A on the dimension n is necessary. Indeed, it was observed in [26] that on the ball B(0,R)Rn+1 the following holds for each k:λk=σk2+n1Rσk. We do not know if the dependence of B on n is necessary in general. However, see Theorem 7 for situations where B does not depend on n.

Inequality (4) also holds under weaker hypotheses on M. Let M=M(n,α,κ,h) be the class of smooth compact Riemannian manifolds of dimension n+1 with nonempty boundary Σ, which satisfy (H1) and

  • (H2′)

    The Ricci curvature satisfies Ricnα on Mh¯.

  • (H3′)

    The mean curvature of Σ satisfies Hκ.

Then we have the following theorem.

Theorem 4

There exists an explicit constant B=B(n,α,κ,h) such that each manifold M in the class M satisfies the following inequality for each kN,σkB+B2+λk.

Quantitative estimates relating individual Steklov eigenvalues to eigenvalues of the tangential Laplacian ΔΣ have been studied in [29], [6], [2], [3], [20], [31], [26], [30]. They are relatively easy to obtain if the manifold M is isometric (or quasi-isometric with some control) to a product near its boundary. See for example [3, Lemma 2.1]. In this context however, it is usually the quotient λk/σk which is controlled.

Theorem 3 is a generalization of the fundamental result of [26], and also of [30]. For bounded Euclidean domains ΩRn+1 with smooth connected boundary Σ=Ω, Provenzano and Stubbe [26] proved a comparison result similar to Theorem 3, with the constants A and B replaced by a constant CΩ depending on the dimension, the maximum of the mean of the absolute values of the principal curvatures on Σ=Ω and the rolling radius of Ω. Their main insight was to use a generalized Pohozaev identity3 in order to compare the Dirichlet energy of a harmonic function uC(Ω) with the L2-norm of its normal derivative along Σ. In [30], Xiong extended the results of [26] to the Riemannian setting under rather stringent hypotheses. Indeed he considered domains Ω with convex boundary in a complete Riemannian manifold X with either nonpositive (KX0) or strictly positive (KX>0) sectional curvature, with some hypotheses on the shape operator (or second fundamental form). These condition in particular implies that the boundary is connected.

Theorem 3 improves these results in several ways. The first strength of Theorem 3 is that we require geometric control of the manifold M only in the neighbourhood Mh¯ of its boundary. Secondly, we are neither assuming the boundary Σ to be connected nor the sectional curvature to have a constant sign. Finally, we state our results and the proof for any compact Riemannian manifold with boundary. This is more general than bounded domains in a complete manifold (see [24] for a discussion of this question).

Remark 5

Let be the number of connected components of Σ. Then λi=0, for i=1,,. Thus, inequality (3) is of interest only for k+1. Furthermore inequality (4) becomes σk2B for k=1,,. There are indeed examples where σ is arbitrarily small. See Section 6.

Remark 6

The inner and exterior boundary of a domain DM are defined to beID:=DintM,ED:=DΣ. The Cheeger and Jammes constant of M are defined to behM=infDVoln(ID)Voln+1(D) and hM=infDVoln(ID)Voln(ED), where both infima are taken over all domains DM such that Voln+1(D)Voln+1(M)2. In [18], Jammes proved thatσ214hMhM. Another useful lower bound can be deduced from Theorem 3 and the Cheeger inequality for Σ, which states thatλ214hΣ2, where hΣ is the Cheeger constant of the closed manifold Σ. Indeed, it follows from inequality (3) and from inequality (7) thatσ212(A+A2+hΣ2). The interest of this lower bound is that it depends only on the geometry of M on and near its boundary Σ. Of course this is useful only when the boundary Σ is connected, in which case hΣ>0. It is easy to construct examples of manifolds where hMhM is arbitrarily small while 12(A+A2+hΣ2) is bounded away from zero, for instance by attaching a Cheeger dumbbell with thin neck to the interior of M using a thin cylinder.

One could also obtain similar lower bounds for σk from the higher order Cheeger inequality for λk [23], [10], which should be compared with the higher order Cheeger type inequality proved in [14]. The lower bounds for σk given in [18], [14] depends on the global geometry of the manifold and not only the geometry of the manifold near the boundary.

In [30], Xiong extended the results of [26] to domains Ω with convex boundary in a complete Riemannian manifold X with either nonpositive (KX0) or strictly positive (KX>0) sectional curvature, with some hypotheses on the shape operator. This leads to explicit values for the constants A and B. This work was enlightening for us, and leads to Theorem 7 and Corollary 8 below.

Let us now discuss various geometric settings where the constants A and B can be improved.

Theorem 7

Let M be a smooth compact manifold of dimension n+1 with nonempty boundary Σ. Let h¯=roll(M). Let λ,κ+>0.

  • (1)

    Suppose that Σ is totally geodesic. If |K|λ2, then (3) and (4) hold withA=nmax{h¯1,2λπ}+λandB=12(1h¯+nλ). Knowing the sign of the sectional curvature K leads to slightly more precise bounds.

    • a)

      If λ2<K<0, then (3) and (4) hold withA=1h¯+λandB=12(1h¯+nλ).

    • b)

      0<K<λ2, then (3) and (4) hold withA=nmax{h¯1,2λπ}andB=12h¯.

  • (2)

    Suppose that Σ is a minimal hypersurface. If Ricλ2n, then (4) holds with B given in part (1).

  • (3)

    If λ2K0 and each principal curvature satisfies λ<κi<κ+, then (3) and (4) hold withA=nmax{κ+,1h¯}andB=12h¯.

  • (4)

    If 0<K<λ2 and each principal curvature satisfies 0<κi<κ+, then (3) and (4) hold withA=nmax{1h¯,λ2+κ+2}andB=12h¯.

We recover the results in [30] as a consequence of Theorem 7.

Corollary 8

[30, Theorem 1] Let M be a domain in a complete Riemannian manifold with boundary Σ. Let λ,κ+>0. Then

  • (1)

    If λ2K0 and each principal curvature of Σ satisfies λ<κi<κ+, then (3) and (4) hold withA=nκ+andB=12κ+.

  • (2)

    If 0<K<λ2 and each principal curvature of Σ satisfies 0<κi<κ+, then (3) and (4) hold withA=nλ2+κ+2andB=12λ2+κ+2.

We refer the reader to Corollary 35 for another situation where A and B have a simpler expression than those in Theorem 7.

Let us conclude with the situation where K0, which is motivated by the Euclidean case from [26].

Theorem 9

Let M be a smooth compact manifold with nonempty boundary Σ. Suppose that M is flat (K0).

  • If κκiκ+, then (3) and (4) hold withA=nmax{h¯1,κ+}+|κ|andB=12(h¯1+n|κ|).

  • If 0κi<κ+, then (3) and (4) hold withA=nmax{h¯1,κ+}andB=12h¯.

  • If κ<κi0, then (3) and (4) hold withA=h¯1+|κ|andB=12(h¯1+n|κ|).

The proof of Theorem 9 is presented in Section 5.5.

One could also consider manifolds whose boundary admits a neighbourhood which is isometric to the Riemannian product [0,L)×Σ for some L>0, in which case one can take 2B=A=1/L. This is discussed in Section 5.6.

The proof of our main result (Theorem 3) is based on the comparison geometry of principal curvatures of hypersurfaces that are parallel to the boundary. This is presented in Section 2 following a review of relevant Jacobi fields and Riccati equations. The Pohozaev identity is used in Section 3 to relate the Dirichlet energy of a harmonic function to the L2-norm of its normal derivative. The proof of Theorem 3 is presented in Section 4. In Section 5 we specialize to various geometrically rigid settings and prove Theorem 7. Here, we give precise values of the constants A,B occurring in the estimates. In order to do this, we need some 1-dimensional calculations for specific Riccati equations, which are treated in an appendix at the end of the paper. In Section 6, we give various examples to illustrate the necessity of the geometric hypothesis occurring in Theorem 3 and Theorem 7.

Section snippets

Preliminaries from Riemannian geometry

Let M be a smooth compact Riemannian manifold of dimension n+1 with boundary Σ. The distance function f:MR to the boundary Σ is given byf(x)=dist(x,Σ). Any s0 that is small enough is a regular value of f, so that the level sets Σs:=f1(s) are submanifolds of M, which are called parallel hypersurfaces. They are the boundary of Ωs:={xM:f(x)s}. For xΣs, the gradient of the distance function f(x), is the inward normal vector to Σs=Ωs:ν(x):=f(x). In particular, for xΣ, ν(x)=f(x)=n(x). The

Pohozaev identity and its application

Let uC(M) be a harmonic function. The main goal of this section is to obtain a quantitative comparison inequality relating the norms ΣuL2(Σ) and unL2(Σ). Here Σu denotes the tangential gradient of a function uH1(Σ) which is the gradient of u on Σ. To achieve this goal, we need the generalized Pohozaev identity for harmonic functions on M.

Lemma 20 Generalized Pohozaev identity

Let F:MTM be a Lipschitz vector field. Let uC(M) with Δu=0 in M. Then0=ΣunF,udVΣ12Σ|u|2F,ndVΣ+12M|u|2divFdVMMuF,udVM, where

Proof of the main result

We are now ready for the proof of the main result. The proof follows the same lines of argument as in [26, Theorem 1.7]. For the sake of completeness we give the proof here.

Proof of Theorem 3

Let us first recall the variational characterizations of the eigenvalues of Dirichlet-to-Neumann operator σk and of the Laplacian λk. For each kN,σk=infVH1(M),dimV=ksupvV,Σv2dVΣ=1M|v|2dVM;λk=infVH1(Σ),dimV=ksupvV,Σv2dVΣ=1Σ|Σv|2dVΣ. Part a. Let (ψi)iNH1(M) be a complete set of eigenfunctions corresponding to

Signed curvature and convexity

In this section we will give precise bounds on the constants A and B defined in (33) and prove Theorem 7. In the situation where the sectional curvature is constrained to have a constant sign and where we impose convexity assumptions on the boundary Σ, we recover and improve the results of Xiong [30].

Our strategy is to estimate the quantities (h˜δ)a(δ), (h˜δ)b(δ) and (h¯δ)μ(δ) which appear in the definitions (29) and (30) of the constants A and B, and thus also in the definitions (33) of A

Examples and remarks

In this section, we discuss the necessity of the hypothesis of Theorem 3 and give different kinds of examples to illustrate this.

Example 37

The condition on the rolling radius is clearly a necessary condition. An easy example, with two boundary components, is given as follows: Take M=Tn×[0,L] where Tn is an n-dimensional flat torus. The sectional curvature of M is 0, the principal curvatures of M=Tn are 0. As L0, the rolling radius tends to 0, and σk0 for all k ([1]). However, λ3(M) is fixed and

Acknowledgement

Part of this work was done while BC was visiting Université Laval. He thanks the personnel from the Département de mathématiques et de statistique for providing good working conditions. BC acknowledges support of Fonds National Suisse de la Recherche Scientifique, requête 200021–163228.

Part of this work was done while AG was visiting Neuchâtel. The support of the Institut de Mathématiques de Neuchâtel is warmly acknowledged. AG acknowledges support of the Natural Sciences and Engineering

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