An isoperimetric inequality for harmonic mean of Steklov eigenvalues in Hyperbolic space

TL;DR

This paper establishes an isoperimetric inequality for the harmonic mean of Steklov eigenvalues in hyperbolic space, extending the approach to Euclidean space and providing new bounds for eigenvalues in geometric domains.

Contribution

It introduces a novel isoperimetric inequality for Steklov eigenvalues in hyperbolic space and adapts the method to Euclidean space, advancing spectral geometry understanding.

Findings

Proved an inequality for hyperbolic space domains.

Derived similar inequality for Euclidean space.

Provides bounds for Steklov eigenvalues based on domain geometry.

Abstract

In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach to prove this result also gives a similar inequality for the first $n$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Euclidean space.