Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity – GR-Analysis-Geometry

This Research Project is devoted to several mathematical aspects of general relativity. Relying on a close collaboration between analysts and geometers, it is aimed at advancing our knowledge of the analytic and geometric properties of Einstein spacetimes, especially when the metrics under consideration have low regularity. The Einstein equations in certain gauges reduce to nonlinear wave equations, and it is natural to solely impose low regularity assumptions on the initial data set when one solves the initial value problem in general relativity. From a purely physical standpoint, also, spacetimes with low regularity play an important role and allow one to encompass (impulsive) gravitational waves and other curvature singularities. In turn, since general relativity is a geometric theory, there is a strong need for a generalization of certain classical geometric concepts and, in certain classes of manifolds with low regularity, to define and investigate various notions of degenerate metrics and generalized curvature (distributional curvature, one-sided curvature bounds, etc.). In this Project, we thus intend to develop interactions between analysts (with expertise on the Einstein equations) and geometers (with expertise on global Lorentzian geometry).

While being motivated by specific physical problems arising in general relativity, this Project will contribute ---beyond the questions under study---, both, to the theory of nonlinear (hyperbolic, elliptic) partial differential equations with low regularity solutions, on one hand, and, on the other hand, to the geometry of semi-Riemannian manifolds with singular structures (including degenerate light-like metrics, polyhedral metrics, spacetimes with singular curvature associated with colliding particles). While enormous progress was made in recent years on our understanding of Einstein spacetimes, many outstanding problems remain open in mathematical general relativity, and many bridges are yet to be built between various mathematical subfields, which is what this Project is aimed at. From both the analysis and the geometric perspectives, an international effort is building up in this direction and by this Project we intend to be part of it. In the past years, many conferences were organized on mathematical general relativity, and members of both teams participated to these worldwide events.

The members of this Project have the expertise to provide, from various standpoints, major contributions to the proposed topics. We will investigate and make progress on the following issues: Well-posedness theory and stability for the Einstein equations; Global causal geometry of spacetimes with low regularity (incompleteness, curvature blow-up); Characteristic initial value problem for the Einstein equations (low regularity at light-cones). Specifically, the Project will be structured around the following five main Topics:
1. Klainerman curvature conjecture. 2. Propagation and interactions of gravitational waves. 3. Global foliation and late-time behavior of spacetimes with low regularity. 4. Cosmological spacetimes and the initial singularity. 5. Geometry of spacetimes with singular curvature. To tackle these problems significantly, one absolutely needs to combine techniques of analysis and of geometry. This collaboration, therefore, will contribute to establish strong connections between geometers and analysts, both in France and abroad.