The characteristic gluing problem of general relativity

The Einstein equations describe the dynamics of space-time in general relativity. It is well-known that — analogous to the case of Maxwell’s equations — initial data for the Einstein equations needs to satisfy constraint equations. One approach to study the rigidity and flexibility of the Einstein equations is by considering gluing problems for initial data. The so-called spacelike gluing problem for initial data on slices of constant time has been intensively studied by Riemannian geometers. In this talk I will present recent work with S. Aretakis and I. Rodnianski, where we introduce the so-called characteristic gluing problem for initial data along light cones (i.e. characteristic hypersurfaces for the Einstein equations). The characteristic gluing problem is fundamentally different from the spacelike problem, and displays novel rigidity and flexibility features. We moreover show how to apply our characteristic gluing to prove gluing constructions for spacelike initial data. Towards the end of the talk, I will discuss future directions.