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

By L. Mari, M. Rigoli, A. G. Setti. American Journal of Mathematics, 2019

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abstract: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 $w$.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ab5134b0e1713bc0124e375e32a2b000cf168ef9