Regularity of solutions of the isoperimetric problem that are close to a smooth manifold

TL;DR

This paper proves that solutions to the isoperimetric problem near a smooth manifold are themselves smooth and close in regularity, using advanced geometric analysis techniques.

Contribution

It establishes regularity and closeness results for isoperimetric solutions near smooth submanifolds, extending to variable metrics and employing diverse geometric tools.

Findings

Solutions are smooth and $C^{2,eta}$-close to the given submanifold.

Results hold even with variable metrics on the manifold.

Utilizes Allard's regularity theorem and Nash's isometric immersion theorem.

Abstract

In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,\alpha}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allard's regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a parametric version due to Gromov.