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 \(v\rightarrow \infty \). This is achieved by the study of a Plateau-type problem with a 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 \(v\rightarrow \infty \), 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.