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Monotonicity of topological entropy along the Ricci flow near a hyperbolic metric

Karen Butt, Alena Erchenko, Tristan Humbert

Abstract

In 2004, Manning showed that the topological entropy of the geodesic flow of a closed surface of non-constant negative curvature is strictly decreasing along the normalized Ricci flow, and he asked if an analogous result holds in higher dimensions for metrics in a neighborhood of a hyperbolic metric. In this paper, we affirmatively answer this question. Namely, we show that the topological entropy of the geodesic flow of a closed Riemannian manifold that carries a hyperbolic metric is indeed strictly decreasing along the normalized Ricci flow starting from a metric of variable negative sectional curvature sufficiently close to the hyperbolic metric.

Monotonicity of topological entropy along the Ricci flow near a hyperbolic metric

Abstract

In 2004, Manning showed that the topological entropy of the geodesic flow of a closed surface of non-constant negative curvature is strictly decreasing along the normalized Ricci flow, and he asked if an analogous result holds in higher dimensions for metrics in a neighborhood of a hyperbolic metric. In this paper, we affirmatively answer this question. Namely, we show that the topological entropy of the geodesic flow of a closed Riemannian manifold that carries a hyperbolic metric is indeed strictly decreasing along the normalized Ricci flow starting from a metric of variable negative sectional curvature sufficiently close to the hyperbolic metric.

Paper Structure

This paper contains 18 sections, 9 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Related results
  3. Stability estimate
  4. Strategy of the proof
  5. Preliminaries
  6. Topological entropy
  7. Symmetric tensors
  8. Spherical harmonics
  9. Action of the generator of the flow
  10. Decomposition of the space of symmetric tensors.
  11. Curvature tensors
  12. Definition and first properties of the functional $\Phi$
  13. Smoothness of the derivative
  14. Hyperbolic metrics are critical points of $\Phi$
  15. Computing the Hessian of $\Phi$ at a hyperbolic metric
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(M^n,g_{\mathrm{hyp}})$ be a closed hyperbolic manifold of dimension $n\geq 3$. There exist $N=N(n)\in \mathbb N_{\ge 2}$ and $\epsilon>0$ such that for any non-hyperbolic metric $g$ satisfying $\|g~-~g_{\mathrm{hyp}}\|_{C^N}<\epsilon$, the topological entropy is strictly decreasing along the n

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: Slice lemma
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 9 more