Abstract
The aim of this thesis is the mathematical study of families of spacetimes satisfying the Einstein's equations of General Relativity. Two methods are used in this context. The first part, consisting of the first three chapters of this work, investigates the geometric properties of the Emparan-Reall and Pomeransky-Senkov families of 5-dimensional spacetimes. We show that they contain a black-hole region, whose event horizon has non-spherical compact cross sections. We construct an analytic extension, and show its maximality and its uniqueness within a natural class in the Emparan-Reall case. We further establish the Carter-Penrose diagram for these extensions, and analyse the structure of the ergosurface of the Pomeransky-Senkov spacetimes. The second part focuses on the study of initial data, solutions of the constraint equations induced by the Einstein's equations. We perform a gluing construction between a given family of inital data sets and initial data of Kerr-Kottler-de Sitter spacetimes, using Corvino's method. On the other hand, we construct 3-dimensional asymptotically hyperbolic metrics which satisfy all the assumptions of the positive mass theorem but the completeness, and which display an energy-momentum vector of arbitry causal type.