The Isoperimetric Inequality for Compact Rank One Symmetric Spaces and Beyond

TL;DR

This paper extends isoperimetric inequalities to compact rank one symmetric spaces using Klartag's needle decomposition, identifying regions that minimize boundary area for given volume.

Contribution

It provides sharp isoperimetric inequalities for CROSS spaces and characterizes the regions achieving equality, expanding understanding beyond classical cases.

Findings

Isoperimetric regions are geodesic balls or tubes around subspaces.

Results apply to real, complex, and quaternionic projective spaces.

Sharp inequalities are established for these symmetric spaces.

Abstract

Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the classical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the real projective space $\mathbb{R}P^n$, we demonstrate that the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{R}P^k\subset\mathbb{R}P^n$. For the complex projective space $\mathbb{C}P^n$, the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{C}P^k\subset\mathbb{C}P^n$. And for the quaternionic projective space, the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{H}P^k\subset\mathbb{H}P^n$.