Rigidez de hipersuperfícies em variedades Riemannianas

Abstract

In this thesis we obtain some rigidity results for hypersurfaces in Riemannian manifolds satisfying certain geometric conditions. In the first part of this work we prove a local splitting theorem for a spacelike 3-manifold M into a spacetime M when M has a weakly outermost and outer area minimizing spherical MOTS which saturates an upper bound for the area. Furthermore, we prove a result that characterizes the extrinsic local geometry of M, where we show that this manifold can be isometrically embedded into a Nariai spacetime with appropriate cosmological constant. More, under a convexity condition, we generalize our splitting theorem to two different cases: the first case to MOTSs of genus greater than one and the second case to MOTSs of high dimension with negative -constant. In the second part of this thesis we prove an infinitesimal rigidity result for a free boundary minimal surface of index one that saturates a certain upper bound for the length of the boundary in a compact Riemannian 3-manifold with boundary. Moreover, under an additional hypothesis, we prove a rigidity theorem for the ambient manifold. We finish the work with extensions of these two last results for stationary stable surfaces.