An upper bound for the first nonzero Steklov eigenvalue

Abstract

Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds ${\mathrm{Sect}}_g\le \kappa$ for $\kappa \le 0$ and ${\mathrm{Ric}}_g\ge (n-1)Kg$ for $K\le 0$. We prove that for any bounded domain $\Omega \subset M^n$ with diameter $d$ and Lipschitz boundary, if ${\Omega }^{\ast }$ is a geodesic ball in the simply connected space form with constant sectional curvature $\kappa$ enclosing the same volume as $\Omega$, then ${\sigma }_1(\Omega )\le C{\sigma }_1({\Omega }^{\ast })$, where ${\sigma }_1(\Omega )$ and ${\sigma }_1({\Omega }^{\ast })$ denote the first nonzero Steklov eigenvalues of $\Omega$ and ${\Omega }^{\ast }$ respectively, and $C=C(n,\kappa ,K,d)$ is an explicit constant. When $\kappa =K$, we have $C=1$ and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.

Comments are welcome