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

Abstract

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 ${\Omega }^{\ast }$ of a set $\Omega$ 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 $\Omega$ has ${\mathcal{C}}^{1,\alpha }$-boundary, the area of $\partial {\Omega }^{\ast }$ is recovered as the limit of the $p$-capacities of $\Omega$, as $p\to 1^{+}$. 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 $3\le n\le 7$.