Some Isoperimetric Comparison Theorems for Convex Bodies in Riemannian Manifolds

We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (M, g) satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Ω. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential point of view. Moreover, by integrating the differential inequality we obtain sharp comparison theorems: not only can we derive an inequality which should be compared with Lévy-Gromov Inequality but we also show that if Ric > nδ on Ω, then the profile of Ω is bounded from above by the profile of the half-space H δ in the simply connected space form with constant sectional curvature δ. As consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of Ω, and for the first non-zero Neumann eigenvalue for the Laplace operator on Ω.