The isoperimetric profile of a compact Riemannian Manifold for small volumes

Abstract

We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,\alpha}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifold, there is a unique family concentric pseudo balls which contains all the pseudo balls $C^{2,\alpha}$-close to small spheres. This allows us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.