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

TL;DR

This paper proves that the topological entropy of geodesic flows decreases monotonically along the normalized Ricci flow for manifolds near a hyperbolic metric, extending Manning's 2004 result to higher dimensions.

Contribution

It establishes that topological entropy decreases along the Ricci flow near hyperbolic metrics in higher-dimensional manifolds, confirming Manning's conjecture.

Findings

Topological entropy decreases along the normalized Ricci flow near hyperbolic metrics.

The result applies to manifolds with variable negative sectional curvature close to hyperbolic.

The decrease is strict for metrics sufficiently close to the 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.