A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917684513341440 |
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| author | Zhou, Shengxuan |
| author_facet | Zhou, Shengxuan |
| contents | In this paper, we show that for any finite subgroup $Γ< O(4)$ acting freely on $\mathbb{S}^3$, there exists a $4$-dimensional complete Riemannian manifold $(M,g)$ with ${\rm Ric}_g \geq 0 $, such that the asymptotic cone of $(M,g)$ is $C(\mathbb{S}_δ^3 /Γ)$ for some $δ= δ(Γ) >0$. This answers a question of Bruè-Pigati-Semola [arXiv:2405.03839] about the topological obstructions of $4$-dimensional non-collapsed tangent cones. Combining this result with a recent work of Bruè-Pigati-Semola [arXiv:2405.03839], one can classify the $4$-dimensional non-collapsed tangent cone in the topological sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone Zhou, Shengxuan Differential Geometry In this paper, we show that for any finite subgroup $Γ< O(4)$ acting freely on $\mathbb{S}^3$, there exists a $4$-dimensional complete Riemannian manifold $(M,g)$ with ${\rm Ric}_g \geq 0 $, such that the asymptotic cone of $(M,g)$ is $C(\mathbb{S}_δ^3 /Γ)$ for some $δ= δ(Γ) >0$. This answers a question of Bruè-Pigati-Semola [arXiv:2405.03839] about the topological obstructions of $4$-dimensional non-collapsed tangent cones. Combining this result with a recent work of Bruè-Pigati-Semola [arXiv:2405.03839], one can classify the $4$-dimensional non-collapsed tangent cone in the topological sense. |
| title | A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.02279 |