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

In this paper we show the existence of weak solutions w:M→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 Gp of Δp. These bounds, stable as p→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.