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.

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We prove the spacetime positive mass theorem in dimensions less than eight. This theorem asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition, the inequality E ≥ |P | holds, where (E, P ) is the ADM energy-momentum vector. Previously, this theorem was only known for spin manifolds [38]. Our approach is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case of this theorem [30,27]. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author [14]. An important part of our proof is to introduce an appropriate substitute for the area functional that is used in the time-symmetric case to single out certain minimal hypersurfaces. We also establish a density theorem of independent interest and use it to reduce the general case of the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.

We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfaces are gained in the process. The techniques developed in this paper are flexible and can be adapted to other non-variational existence problems.

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Abstract. We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau [29] from dimension n = 3 to dimensions 3 ≤ n < 8. This requires us to address several technical difficulties that are not present when n = 3. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those in [29].

The study of stable minimal surfaces in Riemannian 3-manifolds (M, g) with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when (M, g) is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of Schoen: An asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat R 3 .

Let .M; g/ be a compact Riemannian manifold of dimension 3, and let F denote the collection of all embedded surfaces homeomorphic to RP 2 . We study the infimum of the areas of all surfaces in F . This quantity is related to the systole of .M; g/. It makes sense whenever F is nonempty.In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of .M; g/. Moreover, we show that equality holds if and only if .M; g/ is isometric to RP 3 up to scaling.The proof uses the formula for the second variation of area and Hamilton's Ricci flow.