Stability of Tori under Lower Sectional Curvature
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915077499650048 |
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| author | Brue, Elia Naber, Aaron Semola, Daniele |
| author_facet | Brue, Elia Naber, Aaron Semola, Daniele |
| contents | Let $(M^n_i, g_i)\to (X,d_X)$ be a Gromov-Hausdorff converging sequence of Riemannian manifolds with ${\rm Sec}_{g_i} \ge -1$, ${\rm diam}\, (M_i)\le D$, and such that the $M^n_i$ are all homeomorphic to tori $T^n$. Then $X$ is homeomorphic to a $k$-dimensional torus $T^k$ for some $0\leq k\leq n$. This answers a question of Petrunin in the affirmative. We show this result is false is the $M^n_i$ are homeomorphic tori which are only assumed to be Alexandrov spaces. When $n=3$, we prove the same tori stability under the weaker condition ${\rm Ric}_{g_i} \ge -2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03824 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability of Tori under Lower Sectional Curvature Brue, Elia Naber, Aaron Semola, Daniele Differential Geometry Let $(M^n_i, g_i)\to (X,d_X)$ be a Gromov-Hausdorff converging sequence of Riemannian manifolds with ${\rm Sec}_{g_i} \ge -1$, ${\rm diam}\, (M_i)\le D$, and such that the $M^n_i$ are all homeomorphic to tori $T^n$. Then $X$ is homeomorphic to a $k$-dimensional torus $T^k$ for some $0\leq k\leq n$. This answers a question of Petrunin in the affirmative. We show this result is false is the $M^n_i$ are homeomorphic tori which are only assumed to be Alexandrov spaces. When $n=3$, we prove the same tori stability under the weaker condition ${\rm Ric}_{g_i} \ge -2$. |
| title | Stability of Tori under Lower Sectional Curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2307.03824 |