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arXiv · 2507.01674

Conformal Green functions and Yamabe metrics of Sobolev regularity

Rodrigo Avalos·Albachiara Cogo·Andoni Royo Abrego
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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.

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Rodrigo Avalos, Albachiara Cogo, Andoni Royo Abrego. 2025-07-02. Conformal Green functions and Yamabe metrics of Sobolev regularity. https://arxiv.org/abs/2507.01674

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