An upper bound for the first nonzero Steklov eigenvalue

TL;DR

This paper establishes an upper bound for the first nonzero Steklov eigenvalue of a domain in certain Riemannian manifolds, generalizing known inequalities and providing explicit constants depending on curvature bounds.

Contribution

It proves a new upper bound for Steklov eigenvalues in curved spaces, extending the Brock-Weinstock inequality to manifolds with curvature bounds and explicit constants.

Findings

Derived an explicit constant C for the eigenvalue bound

Extended the Brock-Weinstock inequality to curved manifolds

Identified conditions under which geodesic balls maximize the eigenvalue

Abstract

Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\operatorname{Sect}_g\leq \kappa$ for $\kappa\leq 0$ and $\operatorname{Ric}_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded domain $\Omega \subset M^n$ with diameter $d$ and Lipschitz boundary, if $\Omega^*$ is a geodesic ball in the simply connected space form with constant sectional curvature $\kappa$ enclosing the same volume as $\Omega$, then $\sigma_1(\Omega) \leq C \sigma_1(\Omega^*)$, where $\sigma_1(\Omega)$ and $ \sigma_1(\Omega^*)$ denote the first nonzero Steklov eigenvalues of $\Omega$ and $\Omega^*$ 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.