Monotone Quantities for $p$-Harmonic functions and the Sharp $p$-Penrose inequality

Abstract

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $\Sigma$. Fix a number $1 < p < 3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $\Sigma$ in $M$, which was first proved by Xiao using weak inverse mean curvature flow.