Mean curvature

Curvature

Pure mathematics

Ricci curvature

Harmonic mean

Hausdorff space

Inverse

Mathematics

Laplace transform

Flow (mathematics)

Ricci flow

Harnack's inequality

Mathematical analysis

Geometry

Mean curvature flow

Combinatorics

Isoperimetric inequality

American Journal of Mathematics

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 $w$.

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