Monotonicity of the Liouville entropy along the Ricci flow on surfaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914570116792320 |
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| author | Butt, Karen Erchenko, Alena Humbert, Tristan Mitsutani, Daniel |
| author_facet | Butt, Karen Erchenko, Alena Humbert, Tristan Mitsutani, Daniel |
| contents | We show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is eventually strictly increasing along the normalized Ricci flow (NRF). More precisely, we obtain a new expression for the derivative of the Liouville entropy along an arbitrary conformal deformation in dimension 2, and we prove it is positive in the direction of the NRF for 1/6-pinched metrics. This partially answers a question of Manning from 2004. In addition, we show that the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is strictly increasing along the NRF starting from any metric of non-constant negative curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monotonicity of the Liouville entropy along the Ricci flow on surfaces Butt, Karen Erchenko, Alena Humbert, Tristan Mitsutani, Daniel Dynamical Systems Differential Geometry We show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is eventually strictly increasing along the normalized Ricci flow (NRF). More precisely, we obtain a new expression for the derivative of the Liouville entropy along an arbitrary conformal deformation in dimension 2, and we prove it is positive in the direction of the NRF for 1/6-pinched metrics. This partially answers a question of Manning from 2004. In addition, we show that the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is strictly increasing along the NRF starting from any metric of non-constant negative curvature. |
| title | Monotonicity of the Liouville entropy along the Ricci flow on surfaces |
| topic | Dynamical Systems Differential Geometry |
| url | https://arxiv.org/abs/2504.07290 |