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

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.