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In this paper we show the existence of weak solutions $w : M \rightarrow\mathbb{R}$ of the inverse mean curvature flow starting from a relativelycompact set (possibly, a point) on a large class of manifolds satisfying Riccilower bounds. Under natural assumptions, we obtain sharp estimates for thegrowth of $w$ and for the mean curvature of its level sets, that are wellbehaved with respect to Gromov-Hausdorff convergence. The construction followsR. Moser's approximation procedure via the $p$-Laplace equation, and relies onnew gradient and decay estimates for $p$-harmonic capacity potentials, notablyfor the kernel $\mathcal{G}_p$ of $\Delta_p$. These bounds, stable as $p\rightarrow 1$, are achieved by studying fake distances associated to capacitypotentials and Green kernels. We conclude by investigating some basicisoperimetric 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