The Sharp $p$-Penrose Inequality

Liam Mazurowski, Xuan Yao

Published 2023-05-31Version 1

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $\Sigma$. Fix a number $1 < p < 2$. We prove a sharp mass-capacity inequality relating the ADM mass of $M$ with the $p$-capacity of $\Sigma$ in $M$. Equality holds if and only if $M$ is isometric to a spatial Schwarzschild manifold with horizon boundary. This inequality interpolates between the Riemannian Penrose inequality when $p\to 1$ and Bray's mass-capacity inequality for harmonic functions when $p\to 2$. To prove the mass-capacity inequality, we derive monotone quantities for $p$-harmonic functions on asymptotically flat manifolds which become constant on Schwarzschild.