A geometric capacitary inequality for sub-static manifolds with harmonic potentials

TL;DR

This paper establishes a new geometric capacitary inequality for sub-static asymptotically flat manifolds with harmonic potentials, revealing a monotonic family of functions along level-set flows up to a critical threshold.

Contribution

It introduces a one-parameter family of monotone functions related to the capacity and proves a novel inequality analogous to the Riemannian Penrose Inequality.

Findings

Monotonicity of the family $F_{eta}$ up to the threshold $eta=rac{n-2}{n-1}$.

A geometric capacitary inequality involving horizon capacity.

Capacity plays a role similar to ADM mass in the inequality.

Abstract

In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $\{F_{\beta}\}$ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $\beta=\frac{n-2}{n-1}$ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.