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

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.Comment: The proof of the mesoscale flatness criterion has been revised so that it leads to a stronger statement which formally containes both the blow-down and blow-up scale flatness criteria (i.e., Allard's regularity theorem and the Allard--Almgren/Simon's flatness criterion for exterior minimal surfaces); see Remark 1.11 and the new statements of Theorems 1.9 and 2.