Proper solutions of the 1/H-flow and the Green kernel of the p-Laplacian

We show existence and optimal growth estimate for the weak inverse mean curvature flow issuing from a point, on manifolds with certain curvature and isoperimetric conditions. Some of the results are obtained by proving new decay estimates for the Green kernel of the p-Laplacian which fix a gap in the literature. Additionally, we address the convergence of renormalized p-capacitary potentials to the inverse mean curvature flow with outer obstacle.