In this work we provide the full description of the upper levels of the classical causal ladder for spacetimes in the context of Lorenztian length spaces, thus establishing the hierarchy between them. We also show that global hyperbolicity, causal simplicity, causal continuity, stable causality and strong causality are preserved under distance homothetic maps.

By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for sequences of 3-dimensional tori M j that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound, a uniform lower bound on the area of the smallest closed minimal surface, and scalar curvature bounds of the form R gM j ≥ −1/j. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang-Lee, Huang-Lee-Sormani and Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus (M, g M ) is replaced by a bound on the quantity − T min{R gM , 0}dvol gT , where M = graph(f ), f : T → R and (T, g T ) is a flat torus.1 the mean curvature convention is that spheres have positive mean curvature with respect to the inner pointing normal vector A 0 arising from (T i , f i ) and m(f i ) → 0 subconverges in intrinsic flat sense to some integral current space (X ∞ , d X∞ , S X∞ ). By adapting Sormani's Arzela-Ascoli Theorem 8.1 in [Sor14] we obtain an Arzela-Ascoli limit function that can be extended to a covering map p∞ : R 3 → X ∞ . Studying the properties of X ∞ and p∞ we conclude that (X ∞ , d X∞ ) is isometric to T ∞ . Contents

Abstract. Let g be a metric on the 2-sphere S 2 with positive Gaussian curvature and H be a positive constant. Under suitable conditions on (g, H), we construct smooth, asymptotically flat 3-manifolds M with non-negative scalar curvature, with outer-minimizing boundary isometric to (S 2 , g) and having mean curvature H, such that near infinity M is isometric to a spatial Schwarzschild manifold whose mass m can be made arbitrarily close to a constant multiple of the Hawking mass of (S 2 , g, H). Moreover, this constant multiplicative factor depends only on (g, H) and tends to 1 as H tends to 0. The result provides a new upper bound of the Bartnik mass associated to such boundary data.

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model timesymmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant.Following the ideas of Mantoulidis and Schoen [16], of Miao and Xie [20], and of joint work of Miao and the authors [6], we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

We obtain higher dimensional analogues of the results of Mantoulidis and Schoen in [8]. More precisely, we show that (i) any metric g with positive scalar curvature on the 3-sphere S 3 can be realized as the induced metric on the outermost apparent horizon of a 4-dimensional asymptotically flat manifold with nonnegative scalar curvature, whose ADM mass can be arranged to be arbitrarily close to the optimal value specified by the Riemannian Penrose inequality; (ii) any metric g with positive scalar curvature on the n-sphere S n , with n ≥ 4, such that (S n , g) isometrically embeds into R n+1 as a star-shaped hypersurface, can be realized as the induced metric on the outermost apparent horizon of an (n+1)-dimensional asymptotically flat manifold with non-negative scalar curvature, whose ADM mass can be made to be arbitrarily close to the optimal value.