On the $1/H$-flow by $p$-Laplace approximation: new estimates via fake distances under Ricci lower bounds

TL;DR

This paper establishes existence and sharp estimates for inverse mean curvature flow solutions on manifolds with Ricci lower bounds, using $p$-Laplace approximation and fake distances, with implications for geometric analysis.

Contribution

It introduces a novel approximation method for inverse mean curvature flow using $p$-Laplace equations and fake distances, providing new estimates under Ricci curvature conditions.

Findings

Existence of weak solutions to inverse mean curvature flow on Ricci lower bound manifolds.

Sharp growth and curvature estimates for solutions and level sets.

Stable bounds as $p o 1$ achieved via fake distances and Green kernels.

Abstract

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and for the mean curvature of its level sets, that are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the $p$-Laplace equation, and relies on new gradient and decay estimates for $p$-harmonic capacity potentials, notably for the kernel $\mathcal{G}_p$ of $\Delta_p$. These bounds, stable as $p \rightarrow 1$, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of $w$.