The Isoperimetric Inequality for Compact Rank One Symmetric Spaces and Beyond

2017/10/11 by Memarian, Yashar
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1710.03952

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 ℝPⁿ, we demonstrate that the isoperimetric regions are given by either the geodesic balls or tubes around some ℝP^k⊂ℝPⁿ. For the complex projective space ℂPⁿ, the isoperimetric regions are given by either the geodesic balls or tubes around some ℂP^k⊂ℂPⁿ. And for the quaternionic projective space, the isoperimetric regions are given by either the geodesic balls or tubes around some ℍP^k⊂ℍPⁿ.

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