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

• Published in: American journal of mathematics Vol. 144; no. 3; pp. 779 - 849
• Main Authors: Mari, Luciano, Rigoli, Marco, Setti, Alberto G
• Format: Journal Article
• Language: English
• Published: Baltimore Johns Hopkins University Press 01.06.2022
• Subjects: Approximation; Curvature; Estimates; Kernels; Laplace equation; Lower bounds; Mathematical analysis
• ISSN: 0002-9327; 1080-6377; 1080-6377
• DOI: 10.1353/ajm.2022.0016

In this paper we show the existence of weak solutions $w:M\rightarrow\Bbb{R}$ 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, which 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 $\scr{G}_p$ 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 $w$.