The Area of Horizons and the Trapped Region

Abstract

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers, curvature estimates, together with a novel surgery construction for marginal surfaces. These results are applied to characterize the boundary of the trapped region.

Existence, Regularity, and Properties of Generalized Apparent Horizons

“…Such curvature estimates were obtained by the authors from stability and the Gauss-Bonnet theorem in [AM05], and then used for the surgery procedure to derive area bounds for certain 2-dimensional horizons. These area bounds and the estimate on the outer injectivity radius were then applied in [AM07] to show that the boundary of the trapped region of a 3-dimensional initial data set is smooth and embedded. This important result of L. Andersson and J. Metzger is the analogue of Theorem 1.2 for marginally outer trapped surfaces in dimension n = 3.…”

“…In Proposition 4.1 we use the Perron method to prove that given two generalized trapped surfaces, there always exists a stable outer minimizing generalized apparent horizon enclosing both of them. The purpose of this proposition in the proof of Conjecture 1 corresponds to that of Lemma 8 in [KH97] and more specifically to that of Lemma 7.7 in [AM07].…”

“…This is analogous to the construction of the apparent horizon as the boundary of the trapped region in [HE73], [KH97], [HI01], [AM07]. The idea of replacing the total union by one increasing union is contained in [KH97] and has been used in [AM07] to prove existence of a smooth outermost apparent horizon in 3-dimensional data sets. A major technical challenge in [AM07] was to show that the boundary of this increasing union is smooth and embedded.…”

Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces

“…In the paper [AM07], inspired by an idea of Schoen [Sch04], we constructed MOTS in the presence of barrier surfaces by inducing a blow-up of Jang's equation. In this context, Jang's equation [SY81,Jan78] is an equation of prescribed mean curvature for the graph of a function in M × R. For details we refer to Sect.…”

“…Here, outermost means that there is no other MOTS on the outside of . From [AM07] it follows that (M, g, K ) always contains a unique such surface, or does not contain outer trapped surfaces at all, under the assumption that ∂ M is outer untrapped. As stated in Theorem 3.1, we show that there exists a solution f to Jang's equation that actually blows up at , assuming that ∂ M is inner and outer untrapped.…”