Isoperimetric residues and a mesoscale flatness criterion for hypersurfaces with bounded mean curvature

TL;DR

This paper fully characterizes large-volume minimizers in the exterior isoperimetric problem with obstacles, introducing a mesoscale flatness criterion for hypersurfaces with bounded mean curvature to analyze the problem.

Contribution

It provides a complete resolution of the exterior isoperimetric problem with obstacles and introduces a new mesoscale flatness criterion for hypersurfaces with bounded mean curvature.

Findings

Identification of the isoperimetric residue as the key obstacle-dependent term

Development of a mesoscale flatness criterion for hypersurfaces with bounded mean curvature

Resolution of the large volume regime in the exterior isoperimetric problem

Abstract

We obtain a full resolution result for minimizers in the exterior isoperimetric problem with respect to a compact obstacle in the large volume regime $v\to\infty$. This is achieved by the study of a Plateau-type problem with free boundary (both on the compact obstacle and at infinity) which is used to identify the first obstacle-dependent term (called {\it isoperimetric residue}) in the energy expansion, as $v\to\infty$, of the exterior isoperimetric problem. A crucial tool in the analysis of isoperimetric residues is a new mesoscale flatness criterion for hypersurfaces with bounded mean curvature, which we obtain as a development of ideas originating in the theory of minimal surfaces with isolated singularities.