Rigidity of a trace estimate for Steklov eigenvalues
Introduction
Let be a compact Riemannian manifold with nonempty boundary. If the following boundary value problem: has a nontrivial solution, then we call the constant σ a Steklov eigenvalue of . Here ν is the unit outward normal vector field on ∂M. The Rayleigh quotient corresponding to Steklov eigenvalues is
Steklov [15], [9] considered this kind of eigenvalue problems because it is closely related to the frequency of liquid sloshing in a container. It is not hard to see that Steklov eigenvalues are just eigenvalues of the Dirichlet-to-Neumann map that sends Dirichlet boundary data of a harmonic function on M to its Neumann boundary data. Steklov eigenvalues were extensively study in the past decades, because it is deeply related to free boundary minimal submanifolds and conformal geometry in differential geometry ([4], [5]), liquid sloshing in physics and Calderón inverse problem ([3], [16]) in applied mathematics.
Higher order Dirichlet-to-Neumann maps were also considered in literature [1], [7], because they are closely related to inverse problems for the Maxwell equation in electromagnetics. However, the Dirichlet-to-Neumann maps considered in [1], [7] was not suitable for spectral analysis. In 2012, Raulot and Savo [12] introduced a new notion of higher order Dirichlet-to-Neumann maps which is suitable for spectral analysis.
We would also like to mention that discrete versions of the classical Dirichlet-to-Neumann maps and higher order Dirichlet-to-Neuman maps were introduced in [6] and [14] respectively.
The Steklov eigenvalues of a Riemannian manifold can be listed in ascending order counting multiplicity as follows: In [13], by further extending the idea of Raulot-Savo in [11], [12], among the others, we obtained the following trace estimate for Steklov eigenvalues: Theorem 1.1 Let be a compact Riemannian manifold with nonempty boundary and V be the space of parallel exact 1-forms on M. Suppose that . Then
In this paper, we characterize the equality case of (1.3). In summary, combining with Theorem 1.1, we have the following result. Theorem 1.2 Let be a compact Riemannian manifold with nonempty boundary and V be the space of parallel exact 1-forms on M. Suppose that . Then The equality holds if and only if M is a metric product of and a closed manifold F with and where is the first positive eigenvalue for the Laplacian operator on F and which is the first eigenvalue of the following boundary value problem:
Moreover, as a direct corollary of Theorem 1.2, one has Corollary 1.1 Let be a bounded domain with smooth boundary. Then, The equality holds if and only if Ω is a round ball.
The key ingredient in the proof of the rigidity in Theorem 1.2 is the following result about triviality of a flat and totally geodesic Riemannian submersion between two complete Riemannian manifolds with boundary. Here, a Riemanian submersion is said to be flat if the horizontal distribution is integrable, and is said to be totally geodesic if each fibre is totally geodesic (see [17]). Theorem 1.3 Let be a flat and totally geodesic Riemannian submersion between two complete Riemannian manifolds with boundary such that . Suppose that M is simply connected. Then, there is a complete Riemanian manifold F without boundary and an isometry such that .
Note that the result fails when M is not simply connected. For example, let M be the annulus with standard metric and be its universal cover. Then we see that the conclusion fails. Moreover, it is not hard to see that the horizontal and vertical distributions of a flat and totally geodesic submersion are both parallel (see [17]). The assumption implies that normal vectors on must be horizontal. So, by the deRham decompositions for Riemannian manifolds with boundary by the second named author [18], we have the splitting conclusion when is simply connected. Theorem 1.3 just gives us a splitting conclusion by replacing the simply connectedness of by the simply connectedness of M which is more suitable for our application in the proof of Theorem 1.2.
Section snippets
Proofs of main results
Let's first recall the notion of development which will be used the in the proof of Theorem 1.3. Let be a curve in . A curve with is called the development of v, where is the parallel displacement from to along γ. This notion was presented in the language of principal fibre bundle in [8]. A proof of the local existence and uniqueness of developments can be found in [18]. Proof of Theorem 1.3 Let be a fixed point and
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