Monotonicity of the Cheeger constant under Ricci flow on spheres
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917079702044672 |
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| author | Williams, Hollis |
| author_facet | Williams, Hollis |
| contents | We study the behavior of the Cheeger isoperimetric constant under the Ricci flow on compact surfaces. For metrics on a surface diffeomorphic to $S^2$, we show that the Cheeger constant is non-decreasing along the flow. The proof uses evolution identities for parallel curves together with a viscosity formulation of the evolution of $\log h$ which accommodates for the possible switching of minimizing regions. We also give examples of nontrivial Ricci flows on topological $2$-spheres for which the Cheeger constant remains constant, demonstrating that strict monotonicity is not expected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monotonicity of the Cheeger constant under Ricci flow on spheres Williams, Hollis Differential Geometry We study the behavior of the Cheeger isoperimetric constant under the Ricci flow on compact surfaces. For metrics on a surface diffeomorphic to $S^2$, we show that the Cheeger constant is non-decreasing along the flow. The proof uses evolution identities for parallel curves together with a viscosity formulation of the evolution of $\log h$ which accommodates for the possible switching of minimizing regions. We also give examples of nontrivial Ricci flows on topological $2$-spheres for which the Cheeger constant remains constant, demonstrating that strict monotonicity is not expected. |
| title | Monotonicity of the Cheeger constant under Ricci flow on spheres |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2404.13063 |