# A Positive Mass Theorem for Manifolds with Boundary

## TL;DR

This paper establishes a positive mass theorem for asymptotically flat manifolds with boundary, linking geometric properties to harmonic functions and extending previous inequalities related to the Riemannian Penrose conjecture.

## Contribution

It introduces a new positive mass theorem involving boundary mean curvature and Green's functions, generalizing existing inequalities in geometric analysis.

## Key findings

- Proves a positive mass theorem for manifolds with boundary.
- Derives inequalities relating mass and harmonic functions.
- Extends Bray's mass-capacity inequality in the context of the Penrose conjecture.

## Abstract

We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application, we obtain an inequality relating the mass and harmonic functions that generalizes H. Bray's mass-capacity inequality in his proof of the Riemannian Penrose conjecture.