Calculus of Variations and Geometric Measure Theory

L. Benatti - A. Pluda - M. Pozzetta

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

created by pluda on 12 Nov 2024

[BibTeX]

preprint

Inserted: 12 nov 2024

Year: 2024

ArXiv: 2411.06462 PDF

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.