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

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 → ∞. This is achieved by the study of a Plateautype problem with free boundary (both on the compact obstacle and at infinity) which is used to identify the first obstacle-dependent term (called isoperimetric residue) in the energy expansion, as v → ∞, 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.