Paper · Feb 2, 2025

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

Gang Li

math.DGarXiv:2502.00660v3
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Abstract

Let be a -D compact surface with boundary and its interior . We show that for a large class of initial and boundary data, the initial-boundary value problem of the normalized Ricci flow , with prescribed geodesic curvature on , has a unique solution for all , and it converges to the complete hyperbolic metric locally uniformly in . Here the natural condition that causes the main difficulty in the a priori estimates in the corresponding initial-boundary problem of the parabolic equations, for which an auxiliary Cauchy-Dirichlet problem is introduced. We also provide examples of the boundary data which fits well with the natural asymptotic behavior of the geodesic curvature, but the solution to fails to converge to the complete hyperbolic metric.

Comments: A few typos are corrected. Careless mistakes in the argument of asymptotic behavior of geodesic curvatures are corrected. The short Preliminaries is merged into Introduction. Comments are welcome!

Ricci flowSurfaces

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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.