On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929379516350464 |
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| author | Zhu, Xingyu |
| author_facet | Zhu, Xingyu |
| contents | We prove that an open manifold with nonnegative Ricci curvature, linear volume growth and noncollapsed ends always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets given large volumes in the above setting. We also find that under our assumptions, the diameter of the level sets of any Busemann function are uniformly bounded as opposed to a classical result stating that they can have sublinear growth when ends are collapsing. Moreover, some equivalent characterizations of linear volume growth are given. Finally, we construct an example to show that for manifolds in our setting, although their limit spaces at infinity are always cylinders, the cross sections can be nonhomeomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_00640 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature Zhu, Xingyu Differential Geometry 53C21, 53C23 We prove that an open manifold with nonnegative Ricci curvature, linear volume growth and noncollapsed ends always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets given large volumes in the above setting. We also find that under our assumptions, the diameter of the level sets of any Busemann function are uniformly bounded as opposed to a classical result stating that they can have sublinear growth when ends are collapsing. Moreover, some equivalent characterizations of linear volume growth are given. Finally, we construct an example to show that for manifolds in our setting, although their limit spaces at infinity are always cylinders, the cross sections can be nonhomeomorphic. |
| title | On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature |
| topic | Differential Geometry 53C21, 53C23 |
| url | https://arxiv.org/abs/2310.00640 |