Stationary black holes: uniqueness, existence

Black holes belong to the most fascinating predictions of Einstein`s theory of gravitation. There exists strong experimental evidence of their existence in our universe. It is therefore a key problem to exhaustively describe the stationary black hole solutions of the field equations, both in space-time dimension four, as well as in higher dimensions, which naturally occur in many current models in theoretical physics. This is the aim of this project. More precisely, in this project we will attempt to 1. Remove those hypotheses of the existing uniqueness theorems for four dimensional black holes which have no justification, physical or otherwise. This includes in particular a) the hypothesis of analyticity of the metric, b) the hypothesis of the non-vanishing surface gravity, and c) the hypothesis of connectedness of the black hole. 2. Prove the existence of five-dimensional stationary vacuum black holes with two-dimensional isometry group (all known solutions have at least a three dimensional isometry group), or prove that such solutions do not exist.

In spite of having been studied for years, black holes still attract tremendous attention in the physics and astrophysics literature. The recent detection by the Laser Interferometric Gravitational Observatory of gravitational waves emitted by several black holes with unexpected masses fuels further interest in the problem. One of the priorities in studies of Einstein equations, or indeed of the universe, is therefore a better understanding of all aspects of the physics and mathematics of black holes.The discovery by Emparan and Reall of five-dimensional black rings opened up a new world of unexpected five-dimensional black hole solutions. All such solutions constructed until this work possess a three-dimensional isometry group.As part of this project, we have constructed stationary black hole solutions of the higher dimensional Einstein equations without any further symmetries except stationarity in the presence of a negative cosmological constant. This is the first rigorous existence result for such non-static solutions.One of the open problems of the theory of uniqueness of four-dimensional stationary black holes is to classify those black holes which possess horizons with vanishing surface gravity, a quantity which has a key influence on the geometry of space-time in a neighborhood of horizons. A second key result of this project is a detailed description of this geometry for black holes which are not-too-far from Kerr black holes.Recent supernova observations in distant galaxy suggest very strongly that we live in a universe with positive cosmological constant. Another key achievement of this project is a pioneering analysis of the energy of gravitational waves emitted by isolated sources in such universes.