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arXiv · 2504.07290

Monotonicity of the Liouville entropy along the Ricci flow on surfaces

Karen Butt·Alena Erchenko·Tristan Humbert·Daniel Mitsutani
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Abstract

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.

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Karen Butt, Alena Erchenko, Tristan Humbert, Daniel Mitsutani. 2025-04-09. Monotonicity of the Liouville entropy along the Ricci flow on surfaces. https://arxiv.org/abs/2504.07290

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