The isoperimetric profile of a compact Riemannian Manifold for small volumes

TL;DR

This paper investigates the shape of small-volume isoperimetric regions in compact Riemannian manifolds, showing they are close to small spheres and reducing the problem to a finite-dimensional variational problem.

Contribution

It introduces pseudo balls as a new class of submanifolds and demonstrates their role in approximating small isoperimetric regions near points in the manifold.

Findings

Small isoperimetric regions are $C^{2,eta}$-close to small spheres.

Existence of unique concentric pseudo balls near each point.

Reduction of the isoperimetric problem to a finite-dimensional variational problem.

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.