A criterion for topological entropy to decrease under normalised Ricci flow

TL;DR

This paper investigates conditions under which the topological entropy of geodesic flows decreases under normalized Ricci flow, extending Manning's results from surfaces to higher-dimensional negatively curved manifolds.

Contribution

It provides a strong criterion ensuring monotonic decrease of topological entropy during Ricci flow for certain negatively curved metrics in higher dimensions.

Findings

Criterion for entropy decrease established

Applies to conformal classes of constant negative curvature

Extends Manning's 2004 results to higher dimensions

Abstract

In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In this short note, we describe the problem and give a strong criterion under which monotonicity of the topological entropy can be established for a short time. In particular, the criterion applies to metrics of negative sectional curvature which are in the same conformal class as a metric of constant negative sectional curvature.