A Riemannian analogue of the Einstein constraint equations

• Applicant: Dr. Andreas Hermann
• Subject Area: Mathematics
• Term: from 2013 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 236332745

Nearly 100 years after the discovery of Einstein's field equations it is still a challenging problem to construct solutions. One method which has proved to be very successful involves the search for solutions of the so-called Einstein constraint equations on a Riemannian manifold (M,g) of dimension 3. If one has a solution, then one can extend the Riemannian metric g to a Lorentzian metric on a 4-dimensional space-time which satisfies Einstein's field equations. In this project we want to consider the Riemannian analogue of this problem. It has been shown that one obtains special Riemannian metrics known as Einstein metrics, if one can solve the Riemannian analogue of the Einstein constraint equations. There are not many results on the solvability of these equations in the literature and it will be the goal of this project to prove the existence of solutions. Surprisingly we can use many results from spin geometry and the analysis of Dirac operators. Apart from that we want to use various methods for non-linear partial differential equations.

• DFG Programme: Research Fellowships
• International Connection: France
• Host: Professor Dr. Emmanuel Marc Humbert