Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces

TL;DR

This paper establishes sharp isoperimetric inequalities for Neumann Laplacian eigenvalues on bounded domains within rank-1 symmetric spaces, extending previous results from hyperbolic spaces and other symmetric spaces.

Contribution

It generalizes existing inequalities to a broader class of rank-1 symmetric spaces, providing new bounds for Neumann eigenvalues.

Findings

Proved sharp inequalities for Neumann eigenvalues in rank-1 symmetric spaces.

Extended previous hyperbolic space results to more general symmetric spaces.

Unified approach for eigenvalue inequalities in various symmetric geometries.

Abstract

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space [13], and Szeg\"o-Weinberger inequality in rank-1 symmetric spaces obtained by Aithal and Santhanam [1].