Einstein constraints: past, present, and future – EINSTEIN-PPF

The project focuses on the global geometry of Riemannian manifolds satisfying Einstein constraints arising in general relativity. In other words, we are interested in initial data sets, namely, spacelike hypersurfaces in a spacetime satisfying Einstein's field equations. Our main objective is to seek a parametrization of ``all'' such hypersurfaces and describe their global geometric and asymptotic properties, such as their behavior at infinity or in the vicinity of gravitational singularities. Despite old and new advances on the subject, including significant contributions by the members of this Project, the geometric analysis of Einstein's constraint equations remains a collection of dispersed results based on ad-hoc techniques and, in fact, still offers many outstanding and challenging open problems: definition of asymptotic invariants, rigidity properties, structure of singularities, etc. Building upon the most recent and spectacular advances in this field, we will seek a unification of the results and methods. We will revisit the notion of mass which is the most fundamental invariant and plays an important role in general relativity and Riemannian geometry, together with several other fundamental invariants (energy momentum, angular momentum, center of mass. The methods and notions of central interest in this Project will include the Conformal Method, Variational Method, linearized curvature operators, nonlinear elliptic estimates, asymptotic symmetries, spinorial structure, and rigidity properties.