Weinstock inequality in hyperbolic space

TL;DR

This paper proves the Weinstock inequality for the first non-zero Steklov eigenvalue in star-shaped mean convex domains within hyperbolic space, extending known results to higher dimensions.

Contribution

It establishes the Weinstock inequality in hyperbolic space for star-shaped mean convex domains, answering an open question for convex domains when dimension is four or higher.

Findings

Proves Weinstock inequality in hyperbolic space for star-shaped mean convex domains

Confirms the inequality for convex domains in hyperbolic space when n ≥ 4

Addresses open question in spectral geometry

Abstract

In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space $\mathbb{H}^n$ for $n \geq 4$. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [7] for the hyperbolic space $\mathbb{H}^n$ when $n \geq 4$.