Conformal Green functions and Yamabe metrics of Sobolev regularity

TL;DR

This paper solves the Yamabe problem for 3-manifolds with Sobolev regularity metrics by developing elliptic theory for the conformal Laplacian and analyzing its Green function, extending classical results to rough metrics.

Contribution

It introduces a new elliptic theory for the conformal Laplacian on Sobolev regular metrics and establishes existence and regularity results for Green functions in this setting.

Findings

Resolved the Yamabe problem for Sobolev class $W^{2,q}$ metrics with $q > 3$ on 3-manifolds.

Developed elliptic theory for the conformal Laplacian on rough metrics.

Established existence, regularity, and blow-up analysis for Green functions in this context.

Abstract

We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class $W^{2,q}$ with $q > 3$. This requires developing an elliptic theory for the conformal Laplacian for rough metrics and establishing existence, regularity and a delicate blow-up analysis for its Green function. Most of the analytical work is carried out in dimensions $n \geq 3$ and for $W^{2,q}$ Riemannian metrics with $q>\tfrac{n}{2}$ and should be of independent interest.