The essential regularity of singular connections in Geometry
Moritz Reintjes, Blake Temple
TL;DR
The paper tackles the problem of identifying the true (essential) regularity of affine connections at and beyond singularities when defined in an $L^p$-based atlas with $p>n$. It introduces the Regularity Transformation (RT) equations as a computable, coordinate-dependent elliptic system that yields regularizing coordinate transformations, and shows that they implicitly regularize the Riemann curvature, enabling a multi-step atlas lifting to essential regularity. A sharp, curvature-based criterion is established: the essential regularity $ess_{ rak M}( Γ)$ is determined by the relative Sobolev regularity between $\nΓ$ and ${\rm Riem}( Γ)$, in $W^{s,p}$ spaces; an iterative RT procedure constructs an atlas where the connection attains this essential regularity, globally when possible. Together, these results provide a definitive theory for distinguishing removable from essential singularities (including GR shock waves and cusp singularities) and for explicitly lifting removable defects to their essential regularity, thus shaping a geometric notion of intrinsic regularity with broad implications in differential geometry and general relativity.
Abstract
We accomplish three things: (i) We discover the geometric (true) regularity of affine connections, their essential (highest possible) regularity, a geometric property independent of starting atlas. (ii) We give a checkable necessary and sufficient condition for determining whether or not connections are at their essential regularity, based on the relative regularity of the connection and its Riemann curvature. (iii) We introduce a computable procedure for lifting any $L^p$ affine connection in an atlas ($p>n$), to a new atlas in which the connection exhibits its essential regularity. To accomplish this, we prove that the RT-equations, originally designed by the authors to locally lift the regularity of singular connections by one derivative, surprisingly, also induce an implicit hidden regularization of the Riemann curvature, together with a global regularization of transition maps between regularizing coordinate charts. From this we deduce a multi-step regularization of the connection, and construct a new atlas in which the connection exhibits its essential regularity. This paper is a culmination of the theory of the RT-equations which provides a computable iterative procedure for lifting an atlas to a new atlas in which the connection exhibits its essential regularity, applicable to any $L^p$ affine connection defined in a $W^{2,p}$ starting atlas, $p>n$. This provides a definitive theory for determining whether singularities in an $L^p$ affine connection are essential or removable by coordinate transformation, together with an explicit procedure for lifting removable singularities to their essential regularity, both locally and globally, $p>n$. This includes GR shock wave singularities and cusp singularities (continuous metrics with infinite gradients) in General Relativity. The essential regularity is the point where an intrinsic level regularity enters the subject of Geometry.