Jang’s equation and its applications to marginally trapped surfaces

Abstract

Abstract. In this paper we survey some recent advances in the analysis of marginally outer trapped surfaces (MOTS). We begin with a systematic review of results by Schoen and Yau on Jang's equation and its relationship with MOTS. We then explain recent work on the existence, regularity, and properties of MOTS and discuss the consequences for the trapped region. We include an outlook with some directions for future research.

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

“…Furthermore, the works of Andersson, Eichmair, and Metzger [2,10,1] imply that with out loss of generality, the above conjecture may be reduced to a simpler case. Define an outermost time-independent apparent horizon to be a time-independent apparent horizon which is not enclosed by any other.…”

“…Define an outermost time-independent apparent horizon to be a time-independent apparent horizon which is not enclosed by any other. The works of Andersson, Eichmair, and Metzger [2,10,1] imply that given any time-independent outer trapped surface, there always exists an outermost time-independent apparent horizon which encloses it (though not necessarily uniquely).…”

“…However, the second Bianchi identity on a manifold N with metric tensor g N implies that div(G) = 0, where G = Ric N − 1 2 R N g N , Ric N is the Ricci curvature tensor, and R N is the scalar curvature. This geometric identity led Einstein to propose (1) G = 8πT, known as the Einstein equation, since as an added bonus we automatically get a conservation-type property for T , namely div(T ) = 0. Naturally this is a very nice feature of the theory since energy and momentum (the spatial components of the energy vector) are conserved in every day experience.…”

Blowup solutions of Jang’s equation near a spacetime singularity

“…However, we can show that the solution arising from the limiting construction in the proof of Theorem 1 (see [6] for details) must blow up once c is sufficiently close to c crit . This distinction is important because, as was observed in [10] (and will be seen in the proof of Theorem 2), solutions can blow up on surfaces that are not apparent horizons.…”

“…Note that Theorem 1 only yields the existence of one global solution -it does not imply that every solution is global in the absence of apparent horizons. More generally, Schoen and Yau's analysis shows that any solution arising from a limit of suitably regularized boundary-value problems can only blow up on an apparent horizon (see Section 3.5 of [6]). This was used in [7] and [8] to prove the existence of apparent horizons in the presence of suitable geometric barriers.…”