Differentiability of the isoperimetric profile and topology of analytic Riemannian manifolds

TL;DR

This paper investigates the smoothness of the isoperimetric profile in real analytic Riemannian manifolds, revealing that such smooth profiles are rare and typically only occur on topological spheres under certain conditions.

Contribution

It demonstrates that smooth isoperimetric profiles are exceptional and characterizes when they occur, linking differentiability to the topology of the manifold.

Findings

Smooth isoperimetric profiles are rare in real analytic Riemannian manifolds.

Under certain conditions, smooth profiles occur only on topological spheres.

The paper establishes a connection between the differentiability of the profile and the manifold's topology.

Abstract

We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumption, this can happen only on topological spheres.