Low-regularity metrics arise naturally in the realm of geometric PDEs, especially in relation to physical models. It is well known in Riemannian geometry that the components of such metrics have the best regularity, locally, in harmonic coordinates. In this talk, we introduce a novel approach to globalize this idea and study conformal classes of rough Riemannian metrics on closed 3-manifolds. Given a Riemannian metric in the Sobolev 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, which we fully resolve. For Yamabe-positive metrics, the resolution of this problem in low-regularity relies crucially on the existence, regularity and a fine blow-up analysis of the Green function of the conformal Laplacian, which we provide in any dimension $n \geq 3$ and for $q>n/2$. This is based on joint work with R. Avalos and A. Royo Abrego.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006