  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 $w$.
