The Einstein Constraint Equations The Einstein constraint equations (ECE) have their roots in the evolution problem of initial data in general relativity (GR) providing necessary and sufficient conditions for well-posedness. As such, their analysis has become a central area of research within mathematical GR. In particular, their conformal formulation naturally relates them to generalised scalar curvature deformation problems and Yamabe-type equations. This conformal formulation has provided classifications for the set of constant mean curvature (CMC) solutions in different settings, although many problems still remain open, especially in the case of far from CMC solutions. Within this mini-course, the objective is to provide an introduction to this topic, reviewing the origins of the ECE, introducing their conformal formulation, and then focusing on CMC classifications where current open questions will be highlighted. 

We consider asymptotically flat Lorentzian manifolds which are foliated by asymptotically Euclidean time-slices with suitably decaying lapse and shift. Each of these time-slices then comes with ten “charges” (energy, linear and angular momentum, and center of mass); the first four of these are well-known to be well-defined under now standard “optimal decay assumptions” on the metric and second fundamental form of the time-slice. For the last six charges, one typically assumes extra decay assumptions called the Regge—Teitelboim conditions, taking the form of asymptotic parity conditions in “Regge—Teitelboim coordinates”. We will first give an introduction to the charges and their (expected) properties and transformation behavior under the asymptotic Poincaré group of the Lorentzian manifold. Next, we will discuss the relevance of the decay conditions for the transformation behavior; this relies on joint works with Christopher Nerz, with Anna Sakovich, and with Saradha Senthil Velu. We will then give examples of asymptotically Euclidean time-slices which do not possess any Regge--Teitelboim coordinates and explain the consequences of this result for the charges; this is joint work with Melanie Graf and Jan Metzger.

We will show that when a spacetime admits a Killing field (a one-parameter family of local isometries), it is possible to define a Finsler structure in a hypersurface transversal to this Killing field that retains all the information about the causality of the spacetime. If this Killing field is not timelike, then the Finsler structure is not defined in all the directions. In particular, this can be used to study Killing horizons and black holes as well as Cauchy developments, and it allows one to describe all the steps in the Causal ladder. We will also describe how the Finsler invariants are related to the causal invariants of the spacetime.

We will prove that given an n-dimensional integral current space and a 1-Lipschitz map, from this space onto the n-dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, then the map has to be an isometry. We deduce as a consequence the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang--Lee--Sormani. (Joint work with G. Del Nin).