When considering the initial value problem in general relativity, the physically meaningful initial data sets consist of a Riemannian manifold together with a symmetric 2-tensor that satisfy the so-called constraint equations. It is a fundamental problem in mathematical relativity to understand these sets.
The purpose of this project is to study and develop geometric and analytic techniques to construct Riemannian manifolds with scalar curvature constraints, with special emphasis on those satisfying properties motivated by open questions in general relativity.
The main goal is to construct families of non time-symmetric asymptotically flat solutions of the constraint equations, i.e., initial data sets representing isolated systems, and use them to test well-known conjectures in general relativity. These conjectures relate the geometry of this type of solutions to a global invariant representing the total mass of the system.