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

In this paper, we proved that for a bounded Hopf-symmetric domain $\mathrm{\Omega }$ in a noncompact rank one symmetric space $M$, the second Dirichlet eigenvalue ${\lambda }_2(\mathrm{\Omega })\le {\lambda }_2(B_1)$ where $B_1$ is a geodesic ball in $M$ such that ${\lambda }_1(\mathrm{\Omega })={\lambda }_1(B_1)$. This generalizes the work of Ashbaugh & Benguria, Benguria & Linde for bounded domains in constant curvature spaces.