Paper · Oct 26, 2026

Variations of the harmonic entropy and linear drift on negatively curved surfaces

Karen Butt, Alena Erchenko, Tristan Humbert

math.DSarXiv:2610.36354
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Abstract

For closed negatively curved surfaces, we obtain formulas for the derivatives of the harmonic entropy of its geodesic flow and the linear drift of its Brownian motion along conformal deformations. We deduce that the harmonic entropy of the geodesic flow of a closed surface of non-constant negative curvature (resp. linear drift of the Brownian motion on its universal cover) is strictly decreasing (resp. increasing) along the normalized Ricci flow. We also prove that any hyperbolic metric on a surface admits a neighborhood in the space of conformal negatively curved metrics in which the linear drift is greater or equal to the Liouville entropy, with equality if and only if the metric is hyperbolic.

Comments: 12 pages

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.