Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

We prove that the isoperimetric profile of a convex domain $\mathrm{\Omega }$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ 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 $\mathrm{\Omega }$. 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 L'evy-Gromov Inequality but we also show that if $$\text{Ric}$\geq n\delta$on$\Omega$,thentheprofileof$\Omega$ is bounded from above by the profile of the half-space ${\mathbb{H}}_{\delta }^{n+1}$ in the simply connected space form with constant sectional curvature $\delta$. As consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of $\mathrm{\Omega }$, and for the first non-zero Neumann eigenvalue for the Laplace operator on $\mathrm{\Omega }$.