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

TL;DR

This paper explores the properties of outward minimising hulls in Riemannian manifolds, linking geometric measure theory with capacity and isoperimetric inequalities, under curvature assumptions.

Contribution

It characterizes outward minimising hulls as solutions to a minimal surface problem and connects their area to p-capacities, deriving a sharp isoperimetric inequality.

Findings

Outward minimising hulls are maximal volume solutions to a least area obstacle problem.

The boundary area of hulls is obtained as the limit of p-capacities as p approaches 1.

A sharp isoperimetric inequality is established for manifolds with nonnegative Ricci curvature.

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 $\Omega^*$ 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 $\mathscr{C}^{1, \alpha}$-boundary, the area of $\partial \Omega^*$ 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 $3 \leq n \leq 7$.