Global solutions of the vacuum Einstein equations with initial data on a light-cone

The only known systematic method for constructing solutions of the vacuum Einstein equations is provided by the Cauchy problem. The object of this project is to establish local and global existence theorems for the Cauchy problem with initial data provided on a light-cone. The main issues that we plan to address are: 1. The current existence theory for non-linear characteristic initial value problems only provides solutions near the tip of the light-cone, or near the intersection surface. We will prove that a solution exists near a maximal subset of the initial data surface, defined by the transport equations for the first transverse derivatives. 2. We will prove that initial data on the light-cone which are suitably close to those for data for Minkowski space-time lead to solutions which are global to the future of the light-cone. 3. We will construct space-times with initial data given on "past null infinity". More precisely, the initial data will be the incoming radiation fields prescribed in the remote past at large retarded times. Such a problem can be conformally reduced to one where specific gravitational fields are prescribed on a light-cone which has been "pushed infinitely far away to the past".

At the heart of general relativity lie Einsteins field equations which describe in which way the presence of matter curves spacetime. They form a complicated system of partial differential equations for the gravitational field. A key direction of research is the construction of solutions to these equations which represent physically relevant configurations. In particular, one is interested in so-called asymptotically ?at spacetimes where the gravitational field shows a specific fall-off behavior at infinity, which then provide models for isolated gravitational systems.The main aim of this project was to construct such space-times. For this, a new method to construct such spacetimes has been developed, based on a new approach to the characteristic initial value problem, using a new reformulation of the Einstein equations. Detailed mathematical results concerning these equations has been proved, as needed to rigorously prove existence of the associated solutions. We have thus successfully attained our goal, to construct new large classes of asymptotically ?at solutions to Einsteins field equations. The large-distance properties of the solutions have been described in detail.Because solutions of symmetries play a distinguished role in physics, we have extended the initial goals and given an exhaustive description of characteristic initial data sets leading to solutions to Einsteins equations with symmetries. Combining this with our previous analysis, one obtains new families of spacetimes which are both asymptotically ?at and admit symmetries.