Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive Bi-Ricci curvature
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909233984831488 |
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| author | Zhou, Jie Zhu, Jintian |
| author_facet | Zhou, Jie Zhu, Jintian |
| contents | In this paper, we prove the optimal volume growth for complete Riemannian manifolds $(M^n,g)$ with nonnegative Ricci curvature everywhere and bi-Ricci curvature bounded from below by $n-2$ outside a compact set when the dimension is less than eight. This answers a question [AX24, Question 1] proposed by Antonelli-Xu in dimensions six and seven. As a by-product, we also prove an analogy of Gromov's volume bound conjecture [Gro86, Open Question 2.A.(b)] under the condition of positive bi-Ricci curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_18343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive Bi-Ricci curvature Zhou, Jie Zhu, Jintian Differential Geometry In this paper, we prove the optimal volume growth for complete Riemannian manifolds $(M^n,g)$ with nonnegative Ricci curvature everywhere and bi-Ricci curvature bounded from below by $n-2$ outside a compact set when the dimension is less than eight. This answers a question [AX24, Question 1] proposed by Antonelli-Xu in dimensions six and seven. As a by-product, we also prove an analogy of Gromov's volume bound conjecture [Gro86, Open Question 2.A.(b)] under the condition of positive bi-Ricci curvature. |
| title | Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive Bi-Ricci curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.18343 |