On a Conjecture of Ashbaugh and Benguria about Lower Eigenvalues of the Neumann Laplacian

TL;DR

This paper proves a new isoperimetric inequality for lower eigenvalues of the Neumann Laplacian, strengthening existing results and supporting a significant conjecture in spectral geometry.

Contribution

It introduces a strengthened inequality for eigenvalues of the Neumann Laplacian on bounded domains, advancing the understanding of spectral properties in Euclidean and hyperbolic spaces.

Findings

Established a new isoperimetric inequality for lower eigenvalues

Strengthened the Szeg"o-Weinberger inequality

Provided evidence supporting the Ashbaugh-Benguria conjecture

Abstract

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szeg\"o-Weinberger inequality and supports strongly an important conjecture of Ashbaugh-Benguria.