Optimal regularity and Uhlenbeck compactness for general relativity and Yang–Mills theory

We announce the extension of optimal regularity and Uhlenbeck compactness to general setting connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including Lorentzian metric General Relativity. Compactness is essential tool mathematical analysis for establishing validity approximation schemes. Our proofs are based theory RT-equations $L^p$ curvature. Solutions furnish coordinate transformations which give a non-optimal connection gain one derivative its Riemann curvature, (i.e., regularity). The elliptic regardless signature, regularize singularities in solutions hyperbolic Einstein equations. As an application, at GR shock waves removable, implying geodesic curves, locally inertial coordinates Newtonian limit all exist. By extra we extend from Uhlenbeck's compact Riemannian case manifolds. version can also be viewed as "geometric" improvement Div-Curl Lemma, improving weak continuity wedge products strong convergence.