An isoperimetric inequality for the harmonic mean of Steklov eigenvalues in rank one symmetric spaces

TL;DR

This paper establishes an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues on bounded domains within noncompact rank-1 symmetric spaces, extending previous Euclidean and hyperbolic space results.

Contribution

It generalizes known inequalities from Euclidean and hyperbolic spaces to a broader class of noncompact rank-1 symmetric spaces.

Findings

Proves an isoperimetric inequality for Steklov eigenvalues in rank-1 symmetric spaces.

Extends previous results from Euclidean and hyperbolic geometries.

Provides a new geometric inequality relevant to spectral geometry.

Abstract

In this note, an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues is proved on bounded domains in noncompact rank-$1$ symmetric spaces. This work extends result of \cite{BR.01} and \cite{V.21} proved for bounded domains in Euclidean space and Hyperbolic space, respectively.