Uniqueness and stability of hypersurfaces in Semi-Riemannian Spaces.

Abstract

This thesis is divided into two independent parts. In the first one, we study the geometry of immersions of an n-dimensional manifold into semi-Riemannian ambient spaces. These ambient spaces consist in warped products of an open interval of the real line and of an n-dimensional Riemannian manifold (called the fiber), where the warping function is defined on the interval, furnished with a weight funtion that does not deppend on the parameter of the interval. Such an ambient is naturally foliated by means of totally umbilical leaves, called slices, which are isometric to the fiber of the ambient. Endowed with the Riemannian metric induced from the metric tensor of the ambient, the immersed manifolds are also called hypersurfaces (spacelike hypersurfaces when the ambient is a Lorentzian one). The aim of the first part is to study certain sufficient conditions, related to the interaction between the geometries of the ambient and of a given hypersurface and the weight function, to guarantee that the hypersurface is a slice of the ambient. To do so, we apply a variety of analytic tools to the height function and to the angle function of a hypersurface, such as maximum principles, conditions involving the Lp spaces, and criteria of parabolicity. In the second part, we consider the variational problem of minimizing the s-area funtional while keeping constant a functional defined as a linear combination of the r-area functional and the balance of volume. The critical points of this problem are hypersurfaces such that a certain ratio between their symmetric funtions of order r and s (or, equivalently, between their corresponding mean curvatures) is constant, which leads us to the notion of (strong or not) (r, s, a, b)-stability. Under certain reasonable geometric conditions, and assuming that a constant, which appears when we compute the second variation of the Jacobi functional associated with this variational problem, is nonpositive, we show that the geodesic sphere are the only (r, s, a, b)-stable closed hypersurfaces of the space forms and the only strongly (r, s, a, b)-stable closed hypersurfaces of the hyperbolic space, and that the totally umbilical round are the only strongly (r, s, a, b)-stable compact hypersurfaces of the De Sitter space