Rodrigo Avalos (Tübingen) will give the geometry seminar on 9th Feb at 2pm in the Salle des Profs. Rodrigo’s title is “Regularity of compactified AE 3-manifolds and optimal structures of infinity” and his abstract is below.
Asymptotically Euclidean (AE) manifolds play a central role in general relativity, where they serve as models for isolated gravitational systems. These are non-compact manifolds which approach Euclidean space near infinity, and the rate of this convergence is crucial for their total mass, energy-momenta and centre of mass to be well-defined. Despite this, a clear geometric characterisation of conditions guaranteeing sufficiently strong asymptotics making all these quantities well-defined remains an open problem. From a geometric perspective, this issue can be reformulated as the problem of finding geometrically constructed asymptotic coordinates in which the metric exhibits optimal decay properties. In this talk, we relate this problem to a purely geometric one: The optimal regularity of a 1-point conformal compactification of such manifolds. We show how this compactification problem can be addressed via a subtle application of elliptic regularity theory, and how improved regularity of the compactified manifold leads to the existence of geometrically constructed asymptotic coordinates with optimal decay on the original AE manifold.