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.

The formula at the bottom of page 11 in v1 is now corrected in v2 (equation (4.13)). The formula for the derivative of the Liouville measure in v1 is corrected in Lemma 3.1. Some propositions were modified and more details provided but the main theorem statements and the overall idea of the proof are unchanged