I will discuss a classification theorem for metrically complete solutions of the static vacuum Einstein equations with a compact but non-necessarily connected horizon. It is stated that any such solution is either: i) a Boost, ii) a Schwarzschild black hole, or iii) is of Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Korotkin-Nicolai black holes.

I will give an introduction to the differential geometry of surfaces in Thurston's homogeneous 3-manifolds with a particular scope on the role of harmonic map in the theory of constant mean curvature surfaces and minimal surfaces in HxR (H the hyperbolic plane) and the Heisenberg Riemannian space.