The Einstein constraint equations

This course is directed to students and researchers interested in the overlap among Riemannian geometry, geometric analysis and physics. 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 whole area of ​​research within mathematical GR and its intersection with classic problems in geometric analysis has produced plenty of feedback between these areas. The aim of this course is to provide a thorough description of the conformal method, which translates the geometric ECEs into an elliptic system of partial differential equations (PDEs). This subject intersects several traditional problems in geometric analysis, such as scalar curvature prescription and the Yamabe problem.

As prerequisites, classical topics and language both in differential geometry and Riemannian geometry will be assumed. Although functional analysis and PDE experience are beneficial, the course is designed to be self-contained providing necessary background on these topics. The necessary tools on PDE theory and functional analytical methods will be compiled and reviewed in a ready-to-use fashion.

Summarized syllabus:

• Introduction to GR and the Einstein equations

1. The Einstein equations, special solutions and geometric constructions  
   
2. The Cauchy problem for the Einstein equations

• The constraint equations and the conformal method

1.  The conformal method
2. Review of elliptic theory on closed manifolds
3. Constant mean curvature classification on closed manifolds

• Asymptotically Euclidean (AE) initial data

1. AE manifolds and elliptic operators 
2. Constructions of AE initial data sets