Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

TL;DR

This paper establishes a unified framework connecting monotonicity formulas in geometric flows with $p$-capacitary potentials, revealing new insights into curvature functionals and their limits in Riemannian manifolds.

Contribution

It demonstrates that many monotone quantities along inverse mean curvature flow are limits of those along $p$-capacitary potentials, providing a new unified perspective.

Findings

Monotone quantities along inverse mean curvature flow are limits of $p$-capacitary potential level sets.

Strong convergence of $p$-capacitary potentials to inverse mean curvature flow as $p o 1^+$.

A Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials.

Abstract

We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In $3$-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of $p$-capacitary potentials. We prove that they strongly converge in $W^{1,q}_{\mathrm{loc}}$ as $p\to 1^+$ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials.