A Local Method for Compact and Non-compact Yamabe Problems
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910708010057728 |
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| author | Xu, Jie |
| author_facet | Xu, Jie |
| contents | Let $ (M, g) $ be a compact manifold or a complete non-compact manifold without boundary, $ \dim M \geqslant 4 $, and not locally conformally flat. In this article, we introduce a new local method to resolve the Yamabe problem on compact manifold for dimensions at least $ 4 $, and the Yamabe problem on non-compact complete manifolds without boundary, which are pointwise conformal to subsets of some compact manifolds. In particular, the new local method applies to the hard cases--the Yamabe constants are positive. Our local method also generalizes Brezis and Nirenberg's nonlinear eigenvalue problem to subsets of manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13537 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Local Method for Compact and Non-compact Yamabe Problems Xu, Jie Differential Geometry 58J05, 35J60, 53C18 Let $ (M, g) $ be a compact manifold or a complete non-compact manifold without boundary, $ \dim M \geqslant 4 $, and not locally conformally flat. In this article, we introduce a new local method to resolve the Yamabe problem on compact manifold for dimensions at least $ 4 $, and the Yamabe problem on non-compact complete manifolds without boundary, which are pointwise conformal to subsets of some compact manifolds. In particular, the new local method applies to the hard cases--the Yamabe constants are positive. Our local method also generalizes Brezis and Nirenberg's nonlinear eigenvalue problem to subsets of manifolds. |
| title | A Local Method for Compact and Non-compact Yamabe Problems |
| topic | Differential Geometry 58J05, 35J60, 53C18 |
| url | https://arxiv.org/abs/2410.13537 |