Riemannian structures of limited regularity arise naturally in the realm of geometric PDEs. It is well known, since the work of Sabitov-Shefel and De Turck-Kazdan in the late seventies, that the optimal regularity of a Riemannian structure is governed by that of the Ricci tensor in harmonic coordinates.

In this talk, we will discuss the conformal analogue problem. More precisely, we consider a closed three-dimensional Riemannian manifold $(M,g)$ with $g$ in the Sobolev class $W^{2,q}$ with $q>3$ and show that the optimal regularity of the conformal structure is governed by that of the conformally invariant Cotton tensor. The proof requires the resolution of the Yamabe problem for $W^{2,q}$ metrics, which is also interesting on its own.

This is based on a joint work with R. Avalos and A. Royo Abrego.