### Mathematics in Engineering

2022, Issue 2: 1-40. doi: 10.3934/mine.2022013
Research article Special Issues

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

• Received: 09 January 2021 Accepted: 19 May 2021 Published: 02 July 2021
• 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.

Citation: Virginia Agostiniani, Lorenzo Mazzieri, Francesca Oronzio. A geometric capacitary inequality for sub-static manifolds with harmonic potentials[J]. Mathematics in Engineering, 2022, 4(2): 1-40. doi: 10.3934/mine.2022013

### Related Papers:

• 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.

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