Two balls maximize the third Neumann eigenvalue in hyperbolic space

TL;DR

This paper proves that in hyperbolic space, the maximum third Neumann eigenvalue for a given volume is achieved by two disjoint geodesic balls, extending Euclidean results and offering a new proof approach.

Contribution

It extends the maximization result of the third Neumann eigenvalue to hyperbolic space and introduces a novel proof technique.

Findings

Disjoint union of two geodesic balls maximizes the third Neumann eigenvalue

Extends Euclidean space results to hyperbolic geometry

Provides a new proof of the key step in the eigenvalue maximization

Abstract

We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument