Regularity of conformal structures on closed 3-manifolds

TL;DR

This paper investigates the regularity of conformal structures on closed 3-manifolds, linking the existence of more regular representatives in conformal classes to the Yamabe problem for rough metrics.

Contribution

It characterizes when a more regular conformal representative exists for rough metrics on closed 3-manifolds, connecting to the Yamabe problem and applications in special geometric structures.

Findings

Characterization of regular conformal representatives for rough metrics

Connection to the Yamabe problem for low-regularity metrics

Applications to conformally flat, static, and Einstein manifolds

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.