Riemannian structures of limited regularity arise naturally in the realm of geometric PDEs, especially in relation to physical models. Since the work of Sabitov-Shefel and De Turck-Kazdan in the late seventies, it is well known 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 3-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 renowned Yamabe problem for W^{2, q} metrics, which is interesting on its own.In turn, the resolution of the Yamabe problem relies on new results on the existence, regularity and expansion of the Conformal Green's function for W^{2, q} metrics, which we provide in any dimension n greater or equal to 3 and for q>n/2.This is based on a joint work with R. Avalos and A. Royo Abrego.