Sharp isoperimetric inequalities and sphere theorems

Shihshu Walter Wei and Meijun Zhu

Vol. 220 (2005), No. 1, 183–195

DOI: 10.2140/pjm.2005.220.183

Abstract

Various relations between sharp isoperimetric inequalities and volumes of manifolds are studied. In particular, we introduce and estimate sharp isoperimetric constants τ^∗ and γ^∗ corresponding to two types of isoperimetric inequalities. We show that for a complete n-dimensional manifold M with Ricci curvature Ric(M) ≥ n− 1, the volume of M is close to that of Sⁿ if and only if τ^∗ is close to n(n − 1)∕2(n + 2)ω_n^2∕n and M is simply connected (for n = 2 or 3), or γ^∗ is close to 1 (for any n ≥ 2).

Keywords

isoperimetric inequality, Ricci curvature, sectional curvature, Sobolev inequality

Mathematical Subject Classification 2000

Primary: 58E35, 53C20, 53A99

Milestones

Received: 5 February 2004

Revised: 3 August 2004

Published: 1 May 2005