A Volumetric Penrose Inequality for Conformally Flat Manifolds

Abstract

We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to R n \Ω, n ≥ 3, and so that their boundary is a minimal hypersurface. (Here, Ω ⊂ R n is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by 1 2 (V /βn) (n−2)/n , where V is the Euclidean volume of Ω and βn is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999-1016, 2002). Surprisingly, we do not require the boundary to be outermost.

The ADM mass of asymptotically flat hypersurfaces

“…More recently, Bray and Lee [12] established the conjecture for n ≤ 7 with the extra requirement that M be spin for the rigidity statement. Even though many partial results have been obtained ( [23], [38], [22], [27]), the conjecture remains wide open in higher dimensions except for the case of Euclidean graphs recently investigated by Lam [30]. Thus, if (M, g) ⊂ (R n+1 , g 0 ) is an asymptotically flat graph with an inner boundary Γ whose connected components lie on (possibly distinct) horizontal hyperplanes and if we further assume that M is orthogonal to the hyperplanes along Γ, it is proved in [30] that…”

“…Another interesting example of a manifold to which we can attach 'bowl' metrics as in the previous remark appears in the conformally flat case. But notice that, by using the recent lower bound for m h obtained by Freire and Schwartz [38], [22], which of course turns into a lower bound for m h , (1.16) improves to R h ≥ 0 and h is asymptotically flat; see [10].…”

The equality case of the Penrose inequality for asymptotically flat graphs

“…In dimensions less than 8, the inequality was proved by H. Bray and D. Lee, with the extra spin assumption for the equality case [5]. In the case that (M, g) is conformally flat, H. Bray and K. Iga derived new properties of superharmonic functions in R n and proved the Penrose inequality with a suboptimal constant for n = 3 [4], F. Schwartz obtained a lower bound of the ADM mass in terms of the Euclidean volume of the region enclosed by the minimal boundary [25], and J. Jauregui proved a Penrose-like inequality [18]. For the Penrose inequality (with the sharp constant) in dimensions higher than 8, the only result that we know, other than the spherically symmetric case, is the result of G. Lam [19] (cf.…”

Penrose-type inequalities with a Euclidean background

“…Indeed, u is a superharmonic function that approaches one at infinity, with ∂ ν (u) ≤ 0 on ∂M (cf. Lemma 11 of [33]). Strict inequality above holds for the following reason.…”

“…In addition to the work of Bray and Iga [6], who showed a version of the RPI for dimension three, with suboptimal constant, using only properties of superharmonic functions, we also mention other work on Penrose-like inequalities in special cases. Schwartz and Freire-Schwartz proved "volumetric" Penrose inequalities [14,33] for conformally flat manifolds.…”