| Abstract |
I describe joint work with collaborator Moritz Reintjes in which we use the Regularity Transformation Equations (RT-equations) to prove that every connection, regardless of signature or metrical properties, has an inherent level of regularity, what we call its essential regularity, the highest possible regularity achievable by coordinate transformation. It is well known that manifolds by themselves do not have an inherent level of regularity because a low regularity coordinate atlas can transformed to C∞ by mollification, so the essential regularity of a connection is the point at which an invariant level of regularity enters the subject of Geometry. The RT-equations are an elliptic, non-invariant system of equations which determine the Jacobians of coordinate transformations which lift the regularity of a non-Riemannian connection to one derivative above the regularity of its Riemann curvature tensor. The RT-equations have found application to Uhlenbeck compactness, existence and uniqueness for ODEs below the threshold for Picard’s method, and an application to the Strong Cosmic Censorship Conjecture. Our current existence theory for the RT-equations applies to connections with components in Lp, p > n, sufficient to regularize cusp and shock wave singularities in GR, but not yet black hole singularities. |
