Methods and Applications of Analysis

Volume 29 (2022)

Number 4

On the optimal regularity implied by the assumptions of geometry, I: connections on tangent bundles

Pages: 303 – 396



Moritz Reintjes (Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong)

Blake Temple (Department of Mathematics, University of California, Davis, Calif., U.S.A.)


We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection $\Gamma$, with components $\Gamma \in L^{2p}$ and components of its Riemann curvature $\operatorname{Riem}(\Gamma)$ in $L^p$, in some coordinate system, can be smoothed by coordinate transformation to optimal regularity, $\Gamma \in W^{1,p}$ (one derivative smoother than the curvature), $p \gt \max \{ n / 2, 2 \}$, dimension $n \geq 2$. For Lorentzian metrics in General Relativity this implies that 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, and the Einstein equations hold in the strong sense. The proof is based on an $L^p$ existence theory for the Regularity Transformation (RT) equations, a system of elliptic partial differential equations (introduced by the authors) which determine the Jacobians of the regularizing coordinate transformations. Secondly, this existence theory gives the first extension of Uhlenbeck compactness from Riemannian metrics, to general affine connections bounded in $L^\infty$, with curvature in $L^p , p \gt n$, including semi-Riemannian metrics, and Lorentzian metric connections of relativistic Physics. We interpret this as a “geometric” improvement of the generalized Div‑Curl Lemma. Our theory shows that Uhlenbeck compactness and optimal regularity are pure logical consequences of the rule which defines how connections transform from one coordinate system to another—what one could take to be the “starting assumption of geometry”.


General Relativity, shock waves, optimal metric regularity, Uhlenbeck compactness, apparent singularities, Div-Curl lemma, nonlinear elliptic partial differential equations

2010 Mathematics Subject Classification

Primary 83C75. Secondary 58J05.

M. Reintjes is currently supported by CityU Start-up Grant for New Faculty (7200748) and by CityU Strategic Research Grant (7005839); and was supported by the German Research Foundation, DFG grant FR822/10-1, from June 2019 until July 2021; and by FCT/Portugal, (GPSEinstein) PTDC/MAT-ANA/1275/2014 and UID/MAT/04459/2013, from January 2017 until December 2018.

Received 12 September 2022

Accepted 13 March 2023

Published 3 May 2023