Paper · Jan 13, 2026

Path Connectivity of Anosov Metrics on Surfaces

Guilherme Brandão Guglielmo, R. Ruggiero

math.DSarXiv:2601.08656
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Abstract

We construct a class of Riemannian metrics in closed surfaces of genus greater than one, having Anosov geodesic flows, and some regions of positive curvature, such that for each such surface, there exists a smooth curve of conformal deformations that preserves the Anosov property and connects the surface with a Riemannian metric of negative curvature. The conformal deformation does not arise from geometric flows like the Ricci flow, since it is known that such flows might generate conjugate points in the presence of points of positive curvature in the surface.

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Submitted
Jan 13, 2026
Subjects
math.DS, math.DG

Comments: 26 pages, 7 figures

Ricci flowSurfaces

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

    Karen Butt, Alena Erchenko and Tristan Humbert

    Topics Surfaces · Shared terms “conformal deformation”, “geodesic flow”, “negative curvature”

  2. Monotonicity of the Liouville entropy along the Ricci flow on surfaces

    Karen Butt, Alena Erchenko, Tristan Humbert et al.

    Topics Surfaces · Shared terms “conformal deformation”, “negative curvature”, “geodesic flow”

  3. Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

    Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al.

    Topics Surfaces · Shared terms “positive curvature”

  4. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao

    Topics Surfaces

  5. Determinants of pseudo-laplacians and for spinor bundles over Riemann surfaces

    Alexey Kokotov and Dmitrii Korikov

    Topics Surfaces

  6. A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric

    Gang Li

    Topics Surfaces

  7. Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces

    Peter M. Topping and Hao Yin

    Topics Surfaces

  8. Ricci flow with metric torsion on surfaces of positive Euler characteristic

    Shubham Dwivedi

    Topics Surfaces

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.