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arXiv · 1710.03952

The Isoperimetric Inequality for Compact Rank One Symmetric Spaces and Beyond

Yashar Memarian
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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 $\mathbb{H}P^k\subset\mathbb{H}P^n$.

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Yashar Memarian. 2017-10-11. The Isoperimetric Inequality for Compact Rank One Symmetric Spaces and Beyond. https://arxiv.org/abs/1710.03952

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