“Sobolev conformal structures on closed manifolds”
Venerdì 6 Febbraio 2026, ore 12:30 - Aula 2AB45 - Albachiara Cogo (Centro De Giorgi)
Abstract
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 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. In particular, for Yamabe positive metrics, the Yamabe problem in this low-regularity setting requires developing new elliptic theory for the conformal Laplacian, including existence, regularity and a fine blow-up analysis of its Green function, which we provide in any dimension $n \geq 3$ and for $q> \frac n2$.
This is based on joint work with R. Avalos and A. Royo Abrego.
