A Note on Ricci-pinched three-manifolds
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915781983338496 |
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| author | Benatti, Luca Mantegazza, Carlo Oronzio, Francesca Pluda, Alessandra |
| author_facet | Benatti, Luca Mantegazza, Carlo Oronzio, Francesca Pluda, Alessandra |
| contents | Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci--pinching condition $\mathrm{Ric}\geq\varepsilon\mathrm{R} g$ for some $\varepsilon>0$, where $\mathrm{Ric}$ and $\mathrm{R}$ are the Ricci tensor and scalar curvature, respectively. In this short note, we give an alternative proof based on potential theory of the fact that if $(M,g)$ has Euclidean volume growth, then it is flat. Deruelle-Schulze-Simon and Huisken-Körber have already shown this result and together with the contributions by Lott and Lee-Topping led to a proof of the so-called Hamilton's pinching conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05078 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Note on Ricci-pinched three-manifolds Benatti, Luca Mantegazza, Carlo Oronzio, Francesca Pluda, Alessandra Differential Geometry Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci--pinching condition $\mathrm{Ric}\geq\varepsilon\mathrm{R} g$ for some $\varepsilon>0$, where $\mathrm{Ric}$ and $\mathrm{R}$ are the Ricci tensor and scalar curvature, respectively. In this short note, we give an alternative proof based on potential theory of the fact that if $(M,g)$ has Euclidean volume growth, then it is flat. Deruelle-Schulze-Simon and Huisken-Körber have already shown this result and together with the contributions by Lott and Lee-Topping led to a proof of the so-called Hamilton's pinching conjecture. |
| title | A Note on Ricci-pinched three-manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2409.05078 |