How to smooth a crinkled map of space-time: Uhlenbeck compactness for L∞ connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple equations

We present the authors' new theory of the RT-equations ('regularity transformation' or 'Reintjes-Temple' equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Gamma to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem(Gamma). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L-p spaces. The theory and results announced in this paper apply to arbitrary L-infinity connections on the tangent bundle TM of arbitrary manifolds M, including Lorentzian manifolds of general relativity.