On the Riemannian Penrose inequality in dimensions less than eight

Abstract

The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole [HI01]. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique [Bra01]. In this paper we extend the technique of [Bra01] to dimensions less than 8.

On the Geometry of the Level Sets of Bounded Static Potentials

“…When u 0 = 0, such constant coincides with the capacity of the hypersurface ∂M inside (M, g 0 ), which, according to [6], is defined as Cap(∂M, g 0 ) = inf ˆM |Dw| 2 dµ w ∈ Lip loc (M ), w = 0 on ∂M , w → 1 as |x| → +∞ , up to a multiplicative constant. On the other hand, using the asymptotic expansions (1.2) and (1.3) of g 0 and u, the constant value of U 1 can be computed in terms of the ADM mass m > 0 of the static solution as 5) where |S n−1 | denotes the hypersurface area of the unit sphere sitting inside R n . Having this in mind, we introduce, for p ≥ 0 and for a given constant Dirichlet boundary condition u 0 ∈ [0, 1), the functions…”

P. D. E.'s Which Imply the Penrose Conjecture

“…where R is the scalar curvature of (M 3 , g) at each point, and the above traces, norms, and divergences are naturally taken with respect to g and the Levi-Civita connection of g. Then the dominant energy condition on T implies that we must have (6) µ ≥ |J|, which we will call the nonnegative energy density condition on (M 3 , g, k), where again the norm is taken with respect to the metric g on M 3 . Equations 4 and 5 are called the constraint equations because they impose constraints on the Cauchy data (M 3 , g, k) for each initial value problem.…”

“…In fact, the positive mass theorem was proved by SchoenYau in dimensions n ≤ 7 and by Witten in any number of dimensions, but with the additional assumption that M n is spin. The Riemannian Penrose inequality was proved by Bray [3] in dimension 3 using a proof which that author and Dan Lee [6] have generalized to manifolds in dimensions n ≤ 7, and in a slightly weaker form by Huisken-Ilmanen [18] in dimension 3. Since we will be reducing the general case of the Penrose conjecture to the Riemannian Penrose inequality, the techniques presented here have, at a minimum, the potential to address the Penrose conjecture for manifolds with dimensions n ≤ 7.…”

Holographic argument for the Penrose inequality in AdS spacetimes

“…So far we have been assuming four-dimensional spacetimes, but there is no obstruction to considering the Penrose inequality in higher dimensions. The general form of the Penrose inequality in AdS in higher dimensions is given in [5]; a proof of the Riemannian Penrose inequality for asymptotically flat Riemannian manifolds of dimension less than eight is given in [10]. In higher dimensions, however, the Penrose inequality loses its connection with cosmic censorship, since there are unstable black holes in higher dimensions that develop singularities on their horizon when they pinch off, violating cosmic censorship.…”