Solution operators for divergence type equations and relativistic initial data gluing

Abstract: Given two solutions of the Einstein vacuum equation, can youglue them together along a spacelike hypersurface? Since thepioneering work of Corvino and Corvino–Schoen, we know it is possible to glue two initial data on an annulus in the asymptotically flat regime, modulo a 10-parameter obstruction, given by the energy, momentum, center of mass and angular momentum. Recently, Czimek–Rodnianski showed that the 10-parameter obstruction can be removed: instead they only need certain positivity assumptions on the energy-momentum tensor! Their proof of the obstruction-free gluing involves the null-gluing technique developed recently by Aretakis–Czimek–Rodnianski.

We develop a new, simple, spacelike method to obtain the above gluing results, which also optimizes the positivity, regularity and decay assumptions. It is based on solution operators for divergence type equations with nice support properties. I will explain the construction of such solution operators, and how the underlying positivity in the nonlinear part of scalar curvature enters the story. This talk is based on joint work with Sung-Jin Oh and Zhongkai Tao.