Monotonicity of Liouville entropy along the Ricci flow

Abstract: We consider the geodesic flow of a closed negatively curved surface. Its Liouville entropy is an invariant of the measurable dynamics of the flow, which roughly captures the average exponential divergence of nearby trajectories. For negatively curved surfaces of fixed total area, Katok proved this invariant is maximized at hyperbolic metrics, ie, metrics of constant negative curvature. Our main result is that, in this setting, the Liouville entropy is monotonically increasing along the normalized Ricci flow on the space of metrics. This affirmatively answers a question of Manning, and gives a new proof of Katok’s aforementioned result. In addition to geometric and dynamical methods, our proof also uses microlocal analysis. This is joint work with Erchenko, Humbert, and Mitsutani.

differential geometry

Audience: researchers in the topic

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Virtual seminar on geometry with symmetries

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