Rigidity of stable marginally outer trapped surfaces in initial data sets

In this article we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of \textsl{possibly non-compact} marginally outer-trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable marginally outer trapped surface in an initial data set $(M,g,k)$ obeying the dominant energy condition is conformally diffeomorphic to either the plane $\mathbb{C}$ or to the cylinder $\mathbb{A}$ and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly this rules out the existence of trapped regions with cylindrical boundary. In the second part of the article, we restrict our attention to asymptotically flat data $(M,g,k)$ and show that, in that setting, the existence of an unbounded, stable marginally outer trapped surface essentially never occurs unless in a very specific case, since it would force an isometric embedding of $(M,g,k)$ into the Minkowski spacetime as a space-like slice.Comment: Final version, to appear on Ann. Henri Poincar\'e. arXiv admin note: substantial text overlap with arXiv:1310.511

Rigidity of stable minimal hypersurfaces in asymptotically flat spaces

We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. An analogous result holds true up to ambient dimension seven provided polynomial volume growth on the hypersurface is assumed.Comment: Final version, to appear on Calc. Var. Partial Differential Equation

Localizing solutions of the Einstein constraint equations

We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of $N$-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large $T$ we can engineer solutions where any two massive bodies do not interact at all for any time $t\in(0,T)$, in striking contrast with the Newtonian gravity scenario.Comment: Final version, to appear on Inventiones Mathematica

Constrained deformations of positive scalar curvature metrics

We present a series of results concerning the interplay between the scalar curvature of a manifold and the mean curvature of its boundary. In particular, we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. The methods we employ are flexible enough to allow the construction of continuous paths of positive scalar curvature metrics with minimal boundary, and to derive similar conclusions in that context as well. Our work relies on a combination of earlier fundamental contributions by Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao, and on the interplay of Perelman's Ricci flow with surgery and conformal deformation techniques introduced by Cod\'a Marques in dealing with the closed case.Comment: 80 pages; final pre-print version, accepted for publication in JD

Effective versions of the positive mass theorem

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 R. Schoen: An asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat $\mathbb{R}^3$.Comment: All comments welcome! The final version has appeared in Invent. Mat

Existence of Generalized Totally Umbilic 2-Spheres in Perturbed 3-Spheres

It was recently shown by Souam and Toubiana [34] that the (nonconstantly curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless, in this article we show, by perturbative arguments, that all analytic metrics sufficiently close to the round metric g0 on possess generalized totally umbilic 2-spheres, namely critical points of the conformal Willmore functional . The same is true in the smooth setting provided a suitable nondegeneracy condition on the traceless Ricci tensor holds. The proof involves a gluing process of two different finite-dimensional reduction schemes, a sharp asymptotic analysis of the functional on perturbed umbilic spheres of small radius and a quantitative Schur-type Lemma in order to treat the cases when the traceless Ricci tensor of the perturbation is degenerate but not identically zero. For left-invariant metrics on , our result implies the existence of uncountably many distinct Willmore sphere