Rigidity of a trace estimate for Steklov eigenvalues

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Abstract

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Shi and Yu (2016) [13]). Namely, we show that equality of the estimate holds if and only if the manifold is a direct product of a round ball and a closed manifold. The key ingredient in the proof is a splitting theorem for flat and totally geodesic Riemannian submersions which may be of independent interests.

Introduction

Let (Mn,g) be a compact Riemannian manifold with nonempty boundary. If the following boundary value problem:{Δu=0uν=σu has a nontrivial solution, then we call the constant σ a Steklov eigenvalue of (M,g). Here ν is the unit outward normal vector field on ∂M. The Rayleigh quotient corresponding to Steklov eigenvalues isQ(u)=Mu2dVMMu2dVM.

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 (M,g) can be listed in ascending order counting multiplicity as follows:0=σ0<σ1σ2σk. 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 (Mn,g) be a compact Riemannian manifold with nonempty boundary and V be the space of parallel exact 1-forms on M. Suppose that dimV=m>0. Thenσ1+σ2++σmVol(M)Vol(M).

As a direct corollary of Theorem 1.1, we haveσ1+σ2++σnVol(Ω)Vol(Ω) for any bounded smooth domain Ω in Rn. This estimate is sharp because the equality holds when Ω is a round ball. By using the Cauchy-Schwartz inequality, one has1σ1+1σ2++1σnn2Vol(Ω)Vol(Ω). However, this estimate is weaker than Brock's inverse trace estimate [2]:1σ1+1σ2++1σnnVol1n(Ω)Vol1n(Bn) because of the isoperimetric inequality for bounded Euclidean domains.

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 (Mn,g) be a compact Riemannian manifold with nonempty boundary and V be the space of parallel exact 1-forms on M. Suppose that dimV=m>0. Thenσ1+σ2++σmVol(M)Vol(M). The equality holds if and only if M is a metric product of Bm(R) and a closed manifold F with R=mVol(M)Vol(M) andσ(μ1(F))1R where μ1(F) is the first positive eigenvalue for the Laplacian operator on F andσ(μ):=inffC(Bm(R))Bm(R)(f2+μf2)dVBm(R)Bm(R)f2dVBm(R) which is the first eigenvalue of the following boundary value problem:{Δf=μfonBm(R)fν=σfonBm(R).

The restriction (1.8) is necessary in Theorem 1.2. For example, consider M=[1,1]×S1(L) where S1(L) is the round circle with radius L. As computed in [4],σ1(M)=min{σ(1L2),1}=min{1Ltanh(1L),1}. So,σ1(M)=Vol(M)Vol(M)=1 only when 1Ltanh(1L)1.

Moreover, as a direct corollary of Theorem 1.2, one has

Corollary 1.1

Let ΩRn be a bounded domain with smooth boundary. Then,σ1+σ2++σnVol(Ω)Vol(Ω). 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 π:MN 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 π:(Mn+r,g)(Mn,g) be a flat and totally geodesic Riemannian submersion between two complete Riemannian manifolds with boundary such that π(M)=M. Suppose that M is simply connected. Then, there is a complete Riemanian manifold F without boundary and an isometry φ:MF×M such that πMφ=π.

Note that the result fails when M is not simply connected. For example, let M be the annulus {xR2|1x2} with standard metric and M 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 π(M)=M implies that normal vectors on M 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 M is simply connected. Theorem 1.3 just gives us a splitting conclusion by replacing the simply connectedness of M 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 v:[0,T]TpM be a curve in TpM. A curve γ:[0,T]M withγ(t)=P0t(γ)(v(t))andγ(0)=p is called the development of v, where P0t(γ):Tγ(0)MTγ(t)M is the parallel displacement from γ(0) to γ(t) 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 pIntM be a fixed point and F=π1(p)

References (18)

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1

Research partially supported by NSF of China with contract no. 11701355.

2

Research partially supported by NSF of China with contract no. 11571215.

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