Beijing Institute of Mathematical Sciences and Applications

Lars Andersson , Bowen Zhao

Moritz Reintjes

Monday, May 22, 2023 4:00 PM - 5:00 PM

A3-3-301

Zoom 928 682 9093 (BIMSA)

We prove that curvature alone controls the derivatives of a connection, including the gravitational metrics of General Relativity, and the Yang-Mills connections of Particle Physics. Specifically, we prove that the regularity of $L^p$ connections can be lifted by coordinate/gauge transformation to one derivative above their $L^p$ bounded Riemann curvature, (i.e., to optimal regularity), thereby removing apparent singularities in the underlying geometry. This extends the classical result of Kazdan-DeTurck for Riemannian metric connections. As an application to General Relativity, our optimal regularity result implies that the Lipschitz continuous metrics of shock wave solutions of the Einstein-Euler equations are non-singular, (geodesic curves, locally inertial coordinates and the Newtonian limit all exist in a classical sense). By the extra connection derivative, we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry, and from compact to non-compact gauge groups. The proofs are based on our discovery of, and existence theory for, a novel system of non-linear partial differential equations, (the RT-equations), non-invariant equations which are elliptic independent of metric signature, and which provide a general procedure for constructing coordinate and gauge transformations that regularize spacetime connections.