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

Abstract

<abstract><p>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.</p></abstract>

ADM mass, area and capacity in asymptotically flat $3$-manifolds with nonnegative scalar curvature

“…With an argument similar to the one [6], we show here a sharp inequality involving the ratio between the ADM mass and the capacity of the boundary and a sharp upper bound on this latter in terms of the area of the boundary, for asymptotically flat Riemannian 3manifolds with a single end, with a connected, compact boundary and nonnegative scalar curvature, under appropriate assumptions on the topology and on the mean curvature of the boundary. One of the reasons for the interest in these mass-capacity inequalities is to apply them to obtain generalizations of Bunting and Masood-ul-Alam rigidity theorem [19, Theorem 2], as in [39] and [7, Section 5] (based on results in [14,32]).…”

“…In this section we are going to prove our first monotonicity formula. It is a natural version with boundary and for the comparison with the (exterior spatial) Schwarzschild manifold of mass C (see formula (1.4)) of the one shown in [6]. The main difficulty amounts to ensure that the monotonicity survives the critical values of u (solution of Dirichlet problem (1.3)), that, as already recalled in Section 2, form a set of zero Lebesgue measure.…”

“…With the same idea some new monotonicity formulas (even with non-harmonic functions) were then found to study static and sub-static metrics in general relativity [4,7,[11][12][13]. Recently, this level set approach with harmonic functions applied to certain nonparabolic Riemannian 3-manifolds with nonnegative scalar curvature have produced several results, we mention the sharp comparisons about the rate of decay of an energy-like quantity and the area of the level sets of the minimal positive Green function, obtained by Munteanu and Wang in [41] which, as an application, allowed Chodosh and Li [20] to prove a conjecture of Schoen on the stable minimal hypersurfaces in the Euclidean space R 4 and some new proofs of the well-known positive mass theorem [6,16]. Then, considering linearly growing harmonic functions (similarly to [16], which was influenced by a pioneering work of Stern [43]), some asymptotically flat versions of the spacetime positive mass theorem have been proven in [15,31], while the monotonicity formula in [6] has been extended in [3] to the nonlinear potential theoretic setting, replacing the harmonic functions by p-harmonic functions, to obtain a simpler proof of the Riemannian Penrose inequality with a single black hole.…”

Riemannian Penrose inequality via Nonlinear Potential Theory

“…We mention that setting p = 2 and c p = 1 in this formula, one gets the same monotone quantity as the one employed in [5] to provide a Green's function proof of the positive mass theorem.…”

“…In a very recent paper [5], a more sophisticated version of these monotonicity formulas was finally made available for the level sets flow of the Green's functions, in the context of asymptotically flat 3-manifolds with nonnegative scalar curvature, leading to a simple proof of the positive mass theorem (see also [3,6,11,31,39,42] for related results and methods). In the same paper [5, Section 3] a Geroch-type computation was also performed along the smooth level sets flow of p-harmonic functions with nowhere vanishing gradient, to obtain a new proof of the Riemannian Penrose inequality under such (and some other minor) favorable assumption.…”