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

Primary: 58E35, 53C20, 53A99

Authors