Calculus of Variations and Geometric Measure Theory

L. Mari - M. Rigoli - A. Setti

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

created by mari1 on 15 Nov 2020
modified on 30 Aug 2021



Inserted: 15 nov 2020
Last Updated: 30 aug 2021

Journal: Accepted on Amer. J. Math.
Year: 2019

ArXiv: 1905.00216 PDF


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 $w$.