Monotone quantities of $p$-harmonic functions and their applications

TL;DR

This paper introduces monotone quantities for p-harmonic functions on manifolds with nonnegative scalar curvature and applies them to derive inequalities connecting mass, p-capacity, and Willmore functional, including classical limits as p approaches 1.

Contribution

The paper develops new monotonic quantities for p-harmonic functions on curved manifolds and applies them to geometric inequalities involving mass and capacity.

Findings

Derived local and global monotonic quantities for p-harmonic functions.

Established inequalities relating mass, p-capacity, and Willmore functional.

Revealed classical relations as p approaches 1, connecting ADM and Hawking masses.

Abstract

We derive local and global monotonic quantities associated to $p$-harmonic functions on manifolds with nonnegative scalar curvature. As applications, we obtain inequalities relating the mass of asymptotically flat $3$-manifolds, the $p$-capacity and the Willmore functional of the boundary. As $ p \to 1$, one of the results retrieves a classic relation that the ADM mass dominates the Hawking mass if the surface is area outer-minimizing.