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arXiv · 2205.02951

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

Francesco Maggi·Michael Novack
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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.

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Francesco Maggi, Michael Novack. 2022-10-03. Isoperimetric residues and a mesoscale flatness criterion for hypersurfaces with bounded mean curvature. https://arxiv.org/abs/2205.02951

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