The localized seed-to-solution method for the Einstein constraints

04/27/2023 10:30 am - 11:30 am

General Relativity Seminar

Speaker: Philippe G. LeFloch, Sorbonne University and CNRS

Title: The localized seed-to-solution method for the Einstein constraints

Abstract: I will discuss advances on asymptotically Euclidian initial data sets and the variational method introduced by J. Corvino and R. Schoen. This talk is based on joint papers with The-Cang Nguyen (Montpellier) and Bruno Le Floch (Sorbonne Univ. and CNRS). In the vicinity of any given reference data set, we define a “localized seed-to-solution” map, which allows us to parametrize the initial data sets satisfying the Einstein constraints (possibly with matter fields). The parametrization is defined over classes of data sets understood modulo the image of the dual linearized constraints. Our main contribution concerns the sharp behavior of solutions at infinity, which we can arbitrarily localize in asymptotic cones in the sense of A. Carlotto and R. Schoen. Most importantly, as we prove it, the solutions enjoy sharp decay estimates at the harmonic and super-harmonic levels. In the course of this analysis, we discover the notion of ‘asymptotic modulators’, as we call them, or “correctors” to the standard ADM invariants.