Regularity of conformal structures on closed 3-manifolds

TL;DR

This work addresses the problem of upgrading regularity for conformal structures on closed 3-manifolds with rough metrics. By coupling the Yamabe problem for rough metrics with elliptic regularity in harmonic coordinates, the authors prove that if $g \in W^{k,q}(M)$ with $q>3$ and $\mathrm{C}_g \in W^{l,q}(M)$, there exists a conformal representative with constant scalar curvature that lies in $W^{l+3,q}$, thereby clarifying when conformal modifications improve regularity. The Cotton tensor plays a central role, enabling a precise regularity gain and revealing obstructions to further improvement; as consequences, Cotton-flat metrics admit $C^{\infty}$ representatives and static/Einstein geometries admit smooth conformal gauges. The results extend the conformal and Yamabe regularity theory to Sobolev metrics and provide robust, coordinate-invariant tools for geometric PDEs in low-regularity settings, with applications to conformal flatness, static systems, and Einstein metrics.

Abstract

It is well known in Riemannian geometry that the metric components have the best regularity in harmonic coordinates. These can be used to characterize the most regular element in the isometry class of a rough Riemannian metric. In this work, we study the conformal analogue problem on closed 3-manifolds: given a Riemannian metric $g$ of class $W^{2,q}$ with $q > 3$, we characterize when a more regular representative exists in its conformal class. We highlight a deep link to the Yamabe problem for rough metrics and present some immediate applications to conformally flat, static and Einstein manifolds.

Theorems & Definitions (30)

• Theorem 0: Regularity of conformal classes
• Corollary A.1: Regularity of Cotton flat metrics
• Theorem 0: Regularity of constant scalar curvature metrics
• Theorem 0: Regularity of isometry classes
• Corollary B.1: Regularity of static systems
• Corollary B.2: Regularity of Einstein metrics
• Proposition 2.1
• proof
• Theorem 2.3
• proof