On the PPW Conjecture For Hopf-symmetric Sets In Non-compact Rank One Symmetric Space

TL;DR

This paper proves a geometric inequality for the second Dirichlet eigenvalue of Hopf-symmetric domains in noncompact rank one symmetric spaces, extending classical results from constant curvature spaces.

Contribution

It generalizes the PPW conjecture to non-compact rank one symmetric spaces for Hopf-symmetric domains, establishing an eigenvalue bound involving geodesic balls.

Findings

Second eigenvalue bound for Hopf-symmetric domains

Extension of classical PPW conjecture results

Eigenvalue equality characterized by geodesic balls

Abstract

In this paper, we proved that for a bounded Hopf-symmetric domain $\Omega$ in a noncompact rank one symmetric space $M$, the second Dirichlet eigenvalue $\lambda_2 (\Omega) \leq \lambda_2 (B_1)$ where $B_1$ is a geodesic ball in $M$ such that $\lambda_1 (\Omega) =\lambda_1 (B_1)$. This generalizes the work of Ashbaugh & Benguria, Benguria & Linde for bounded domains in constant curvature spaces.