Conformal Green functions and Yamabe metrics of Sobolev regularity
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| Format: | Preprint |
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2025
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| _version_ | 1866912461167263744 |
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| author | Avalos, Rodrigo Cogo, Albachiara Abrego, Andoni Royo |
| author_facet | Avalos, Rodrigo Cogo, Albachiara Abrego, Andoni Royo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_01674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conformal Green functions and Yamabe metrics of Sobolev regularity Avalos, Rodrigo Cogo, Albachiara Abrego, Andoni Royo Analysis of PDEs Differential Geometry 53C18, 58J05, 35J61, 35J20, 35J08 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. |
| title | Conformal Green functions and Yamabe metrics of Sobolev regularity |
| topic | Analysis of PDEs Differential Geometry 53C18, 58J05, 35J61, 35J20, 35J08 |
| url | https://arxiv.org/abs/2507.01674 |