A geometric capacitary inequality for sub-static manifolds with harmonic potentials

In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family {Fβ} of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold β = n/n2/1 and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.