# Capacity inequalities and rigidity of cornered/conical manifolds

## TL;DR

This paper establishes new geometric inequalities relating mean curvature, mass, and boundary conditions of hypersurfaces in convex cones and asymptotically flat manifolds, using inverse mean curvature flow techniques.

## Contribution

It introduces novel capacity inequalities and extends classical geometric inequalities to manifolds with boundary in convex cones.

## Key findings

- Proved capacity inequalities involving mean curvature and mass.
- Derived analogues of classical inequalities like P"olia-Szeg"o, Alexandrov-Fenchel, and Penrose.
- Applied inverse mean curvature flow to hypersurfaces with boundary.

## Abstract

We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of P\"olia-Szeg\"o, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.