The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Introduction
Let M be a smooth compact Riemannian manifold of dimension , with nonempty boundary Σ. The Steklov eigenvalue problem on M is to find all numbers for which there exists a nonzero function which satisfies Here n is the outward unit normal along Σ and is the Laplace-Beltrami operator acting on . The Steklov problem has a discrete unbounded spectrum where each eigenvalue is repeated according to its multiplicity. For background on this problem, see [12], [21] and references therein.
Let be the harmonic extension operator: the function satisfies on Σ and in M. The Steklov eigenvalues of M are the eigenvalues of the Dirichlet-to-Neumann (DtN) map , which is defined by The DtN map is a first order self-adjoint elliptic pseudodifferential operator [28, pages 37–38]. Its principal symbol is given by , while that of the Laplace operator is . Standard elliptic theory [16], [27] then implies that where is the volume of the unit ball . It follows that manifolds and which have isometric boundaries satisfy as .
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 and two compact Riemannian manifolds and with the same boundary such that . If the metrics and have the same Taylor series on Σ, then and have asymptotically equivalent1 Steklov spectra: See [11, Lemma 2.1]. In particular, .
The asymptotic behaviour described by (1) or (2) does not contain any information regarding an individual eigenvalue . In order to obtain such information, stronger hypothesis are needed. Indeed on any smooth compact Riemannian manifold with boundary, there exists a family of Riemannian metrics such that on a neighbourhood of , while for each , See [2] and [5] for two such constructions. If , there also exists a family of Riemannian metrics such that on a neighbourhood of Σ and See [2] for the construction of these families. In this last construction, the neighbourhoods are shrinking to Σ as : However, if the manifolds and are uniformly isometric near their boundary, the situation is completely different.
Theorem 1 Given two compact Riemannian manifolds with boundary and such that their respective boundaries and admit neighbourhoods and which are isometric, there exists a constant C, which depends explicitly on the geometry of , such that for each .
Not only the difference is bounded for each given , 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 ℓ be the number of connected components of the boundary Σ. Under the hypothesis of Theorem 1, it is well known that for the following holds: This was used for instance in [2], [6]. Theorem 1 is stronger since it implies It is also known that , where refer to the mixed Neumann-Steklov problem on . 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 on its boundary Σ, which are denoted . Let and let and be such that and . Consider the class of smooth compact Riemannian manifolds M of dimension with nonempty boundary Σ, which satisfy the following hypotheses:
- (H1)
The rolling radius2 of M satisfies .
- (H2)
The sectional curvature K satisfies on the tubular neighbourhood
- (H3)
The principal curvatures of the boundary Σ satisfy .
Theorem 3
There exist explicit constants and such that each manifold M in the class satisfies the following inequalities for each , In particular, for each , .
Under the hypotheses of Theorem 3 one can take where is a positive constant depending on , β, and . 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 the following holds for each k: 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 be the class of smooth compact Riemannian manifolds of dimension with nonempty boundary Σ, which satisfy (H1) and
- (H2′)
The Ricci curvature satisfies on .
- (H3′)
The mean curvature of Σ satisfies .
Theorem 4
There exists an explicit constant such that each manifold M in the class satisfies the following inequality for each ,
Quantitative estimates relating individual Steklov eigenvalues to eigenvalues of the tangential Laplacian 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 which is controlled.
Theorem 3 is a generalization of the fundamental result of [26], and also of [30]. For bounded Euclidean domains with smooth connected boundary , Provenzano and Stubbe [26] proved a comparison result similar to Theorem 3, with the constants A and B replaced by a constant depending on the dimension, the maximum of the mean of the absolute values of the principal curvatures on 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 with the -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 () or strictly positive () 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 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 ℓ be the number of connected components of Σ. Then , for . Thus, inequality (3) is of interest only for . Furthermore inequality (4) becomes for . There are indeed examples where is arbitrarily small. See Section 6.
Remark 6 The inner and exterior boundary of a domain are defined to be The Cheeger and Jammes constant of M are defined to be where both infima are taken over all domains such that . In [18], Jammes proved that Another useful lower bound can be deduced from Theorem 3 and the Cheeger inequality for Σ, which states that where is the Cheeger constant of the closed manifold Σ. Indeed, it follows from inequality (3) and from inequality (7) that 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 . It is easy to construct examples of manifolds where is arbitrarily small while 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 from the higher order Cheeger inequality for [23], [10], which should be compared with the higher order Cheeger type inequality proved in [14]. The lower bounds for 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 () or strictly positive () 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 with nonempty boundary Σ. Let . Let . Suppose that Σ is totally geodesic. If , then (3) and (4) hold with Knowing the sign of the sectional curvature K leads to slightly more precise bounds. If , then (3) and (4) hold with , then (3) and (4) hold with
Suppose that Σ is a minimal hypersurface. If , then (4) holds with B given in part (1).
If and each principal curvature satisfies , then (3) and (4) hold with
If and each principal curvature satisfies , then (3) and (4) hold with
Corollary 8
[30, Theorem 1] Let M be a domain in a complete Riemannian manifold with boundary Σ. Let . Then
- (1)
If and each principal curvature of Σ satisfies , then (3) and (4) hold with
- (2)
If and each principal curvature of Σ satisfies , then (3) and (4) hold with
Let us conclude with the situation where , 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 (). If , then (3) and (4) hold with If , then (3) and (4) hold with If , then (3) and (4) hold with
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 for some , in which case one can take . 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 -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 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 with boundary Σ. The distance function to the boundary Σ is given by Any that is small enough is a regular value of f, so that the level sets are submanifolds of M, which are called parallel hypersurfaces. They are the boundary of . For , the gradient of the distance function , is the inward normal vector to : In particular, for , . The
Pohozaev identity and its application
Let be a harmonic function. The main goal of this section is to obtain a quantitative comparison inequality relating the norms and . Here denotes the tangential gradient of a function 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 be a Lipschitz vector field. Let with in M. Then 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 and of the Laplacian . For each , Part a. Let 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 , and which appear in the definitions (29) and (30) of the constants and , 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 where is an n-dimensional flat torus. The sectional curvature of M is 0, the principal curvatures of are 0. As , the rolling radius tends to 0, and for all k ([1]). However, 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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