Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

摘要

The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull 12* of a set 12 is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where 12 has C1,alpha-boundary, the area of partial differential 12* is recovered as the limit of the p-capacities of 12, as p -> 1+. Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided 3 < n < 7. (c) 2022 Elsevier Inc. All rights reserved.