Calculus of Variations and Geometric Measure Theory

V. Agostiniani - L. Mazzieri - F. Oronzio

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

created by oronzio on 24 May 2022


Published Paper

Inserted: 24 may 2022
Last Updated: 24 may 2022

Journal: Mathematics in Engineering
Volume: 4
Number: 2
Pages: 40
Year: 2021
Doi: 10.3934/mine.2022013


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 $\{F_{\beta}\}$ of functions which are monotone along the level--set flow of the potential. Such monotonicity holds up to the optimal threshold $\beta=\frac{n-2}{n-1}$ 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.