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Monotonicity of the Liouville entropy along the Ricci flow on surfaces

Karen Butt, Alena Erchenko, Tristan Humbert, Daniel Mitsutani

Abstract

Using geometric and microlocal methods, we show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is strictly increasing along the normalized Ricci flow. This affirmatively answers a question of Manning from 2004. More generally, we obtain an explicit formula for the derivative of the Liouville entropy along arbitrary area-preserving conformal perturbations in this setting. In addition, we show the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is also strictly increasing along the normalized Ricci flow.

Monotonicity of the Liouville entropy along the Ricci flow on surfaces

Abstract

Using geometric and microlocal methods, we show that the Liouville entropy of the geodesic flow of a closed surface of non-constant negative curvature is strictly increasing along the normalized Ricci flow. This affirmatively answers a question of Manning from 2004. More generally, we obtain an explicit formula for the derivative of the Liouville entropy along arbitrary area-preserving conformal perturbations in this setting. In addition, we show the mean root curvature, a purely geometric quantity which is a lower bound for the Liouville entropy, is also strictly increasing along the normalized Ricci flow.

Paper Structure

This paper contains 17 sections, 23 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of surfaces
  4. Geodesic flow and Liouville measure
  5. Horizontal and vertical spaces
  6. The Anosov property and (un)-stable manifolds
  7. Stable manifolds and horocycles
  8. The stable vector field
  9. Liouville entropy
  10. A Jensen-type inequality
  11. Monotonicity of the mean root curvature
  12. Monotonicity of the Liouville entropy
  13. Using the identification $SM\cong M_0\times \partial \tilde{M}$
  14. Using an integration by parts formula
  15. Using Pollicott--Ruelle resonances
  16. ...and 2 more sections

Key Result

Theorem 1

Let $M$ be a smooth closed orientable surface of negative Euler characteristic. Let $g_0$ be a smooth Riemannian metric on $M$ of non-constant negative Gaussian curvature. Let $\varepsilon \mapsto g_{\varepsilon}$ denote the normalized Ricci flow starting from $g_0$. Then

Theorems & Definitions (51)

  • Theorem 1
  • Corollary 2: Corollary 2.5 in Ka
  • Theorem 3
  • Theorem 4
  • Remark 1.1
  • Proposition 2.1: Anosov
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 41 more