Quantitative marked length spectrum rigidity
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918228680245248 |
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| author | Butt, Karen |
| author_facet | Butt, Karen |
| contents | We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$. Using the methods of Hamenstädt, we show the volumes of $M$ and $N$ are approximately equal. We then show the Besson-Courtois-Gallot map $F: M \to N$ is a diffeomorphism with derivative bounds close to 1 and depending on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenstädt and Besson-Courtois-Gallot, which show $M$ and $N$ are isometric if their marked length spectra are equal. We also prove a similar result for compact negatively curved surfaces using the methods of Otal together with a version of the Gromov compactness theorem due to Pugh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_12128 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantitative marked length spectrum rigidity Butt, Karen Differential Geometry Dynamical Systems We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$. Using the methods of Hamenstädt, we show the volumes of $M$ and $N$ are approximately equal. We then show the Besson-Courtois-Gallot map $F: M \to N$ is a diffeomorphism with derivative bounds close to 1 and depending on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenstädt and Besson-Courtois-Gallot, which show $M$ and $N$ are isometric if their marked length spectra are equal. We also prove a similar result for compact negatively curved surfaces using the methods of Otal together with a version of the Gromov compactness theorem due to Pugh. |
| title | Quantitative marked length spectrum rigidity |
| topic | Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2203.12128 |