Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds

Mikhail Karpukhin

doi:10.3934/era.2017.24.011

Abstract

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound for Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.

Highlights

Introduction and main resultsLet (M, g) be a smooth compact Riemannian manifold with smooth boundary
The first is the usual Laplace-Beltrami operator ∆ = ∆∂M acting on C∞(∂M ) with respect to the induced metric g|∂M
Since ∂M is compact, the spectrum of ∆ is discrete and consists of eigenvalues which we denote by 0 = λ1 λ2 λ3 . . . , where eigenvalues are counted with multiplicities

Summary

Introduction and main results

Let (M, g) be a smooth compact Riemannian manifold with smooth boundary. We consider two elliptic self-adjoint operators defined on ∂M. Putting all these numbers together, one obtains an equality in (1). Both those eigenvalues have multiplicity n, we have that inequality (1) is sharp for all m n + 1. As it follows from the remark after Theorem 3.2, the equality in inequality (1) can only be achieved on a ball, i. Let us mention a remarkable series of papers by Fraser and Schoen [3, 4], where the authors investigate the optimal upper bound of the form (3) for p = 1. In the last section we compare our results to a similar result due to Wang and Xia [16]

Background in differential geometry

Dirichlet-to-Neumann operator on differential forms

Non-orientable manifolds