Georgia Institute of Technology College of Sciences

I will discuss a nonlinear elliptic system of partial differential equations arising in Riemannian geometry and General Relativity. Specifically, I will present recent advances on the analysis of asymptotically Euclidean, initial data sets for Einstein’s field equations. In collaboration with Bruno Le Floch (Sorbonne University) I proved that solutions to the Einstein constraints can be glued together along possibly nested conical domains. The constructed solutions may have arbitrarily low decay at infinity, while enjoying (super-)harmonic estimates within possibly narrow cones at infinity. Importantly, our localized seed-to-solution method, as we call it, leads to a proof of a conjecture by Alessandro Carlotto and Richard Schoen on the localization problem at infinity, and generalize P. LeFloch and Nguyen’s theorem on the asymptotic localization problem. This lecture will be based on https://arxiv.org/abs/2312.17706