Isoperimetry with upper mean curvature bounds and sharp stability estimates

TL;DR

This paper establishes sharp stability estimates for boundaries with upper mean curvature bounds, extending Almgren's isoperimetric principle, and characterizes nearly constant mean curvature boundaries under perimeter constraints.

Contribution

It provides the first sharp stability estimates for Almgren's isoperimetric principle with upper mean curvature bounds and describes boundaries with nearly constant mean curvature.

Findings

Sharp stability estimates for boundaries with upper mean curvature bounds.

Characterization of boundaries with almost constant mean curvature under perimeter constraints.

Extension of Almgren's isoperimetric principle to stability and near-constant curvature cases.

Abstract

It was proved by Almgren that among boundaries whose mean curvature is bounded from above, perimeter is uniquely minimized by balls. We obtain sharp stability estimates for Almgren's isoperimetric principle and, as an application, we deduce a sharp description of boundaries with almost constant mean curvature under a total perimeter bound which prevents bubbling.