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, 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 G(p) of Delta(p). 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.