Elliptic problems in General Relativity

The aim of Einstein`s theory of gravity is to obtain a detailed understanding of all physically relevant space-times satisfying the Einstein equations. Mathematical studies of these equations belong to the domain of global geometric analysis. One is thus lead to many beautiful and challenging mathematical problems. The recent direct detection of gravitational waves makes it urgent to gain a deeper insight into several aspects of the theory. One set of questions that arises is the rigorous construction of solutions of the Einstein equations with important physical features, as needed to justify the numerical simulations that are used to interpret the gravitational wave signals observed in the detectors. Another issue is that of understanding of the nature and properties of black hole solutions. Indeed, the gravitational wave observed in September 2015 was emitted by a system of two black holes orbiting around each other, which collided to form a single one; this gave us the first proof of existence of nearby multiple black-hole systems in the universe. After a highly dynamical and extremely violent phase, where the energy equivalent of a total annihilation of three of our suns was emitted in a fraction of a second in form of gravitational waves, the final black hole settled down to a stationary one. A rigorous classification of such solutions is another key to a correct interpretation of the signals observed by the detectors. The aim of this project is to address the above issues. This is done by using methods of the mathematical theory of elliptic partial differential equations. While the questions we address have been open for years, the new techniques recently introduced in the theory of elliptic equations will allow us to settle some of the key problems that remain, within a rigorous mathematical approach to the problems at hand.

The energy is one of the fundamental quantities in physics. Energy exchange is what makes cars mobile, apartments warm, or mobile telephones communicate with each other. A proper understanding of energy is therefore one of the basic insights in any physical theory. It has to be admitted that we still do not understand well the notion of energy in general relativity. This is closely related to the basic principle of general relativity, that free fall switches-off the gravitational forces. As a result, a local notion of energy of the gravitational field remains elusive. Fortunately we have a good notion of total energy of isolated gravitating systems. One of the key questions in this context is, whether or not this total energy is bounded from below. The most important result obtained within this project is the proof that the total energy for gravitating systems with a specific behaviour, namely "spherical conformal infinity", is positive. Various partial results to this effect were known before, but a complete proof without any restrictive assumptions has been given for the first time in our work. Somewhat surprisingly, we have also shown that there exists a large class of gravitating systems which have negative energy, namely those which have a "complicated topology at infinity". No such systems with negative energy were known so far. This part of our study provides yet another illustration of various subtleties which arise when gravitational fields are present.