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

TL;DR

This paper introduces new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds, leading to a sharp mass-capacity inequality related to the p-Penrose conjecture, extending previous work by Xiao.

Contribution

It develops novel monotonicity formulas for p-harmonic functions that are constant on Schwarzschild manifolds, providing a new proof of a sharp mass-capacity estimate.

Findings

Established monotone quantities for p-harmonic functions

Derived a sharp mass-capacity inequality for asymptotically flat manifolds

Connected the results to the 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.