A sharp estimate for Neumann eigenvalues of the Laplace-Beltrami operator for domains in a hemisphere

TL;DR

This paper establishes an isoperimetric inequality for the harmonic mean of the first N-1 non-trivial Neumann eigenvalues of the Laplace-Beltrami operator in domains within a hemisphere, advancing understanding of spectral geometry on spheres.

Contribution

It provides a sharp estimate for Neumann eigenvalues on spherical domains, specifically in hemispherical regions, which was previously unaddressed.

Findings

Proves an isoperimetric inequality for Neumann eigenvalues

Establishes bounds for harmonic mean of eigenvalues in hemispherical domains

Advances spectral geometry understanding on spherical surfaces

Abstract

Here we prove an isoperimetric inequality for the harmonic mean of the first $N-1$ non-trivial Neumann eigenvalues of the Laplace-Beltrami operator for domains contained in a hemisphere of $\mathbb{S}^N$.