Paper · Jun 25, 2016

Rate of curvature decay for the contracting cusp Ricci flow

Peter M. Topping, Hao Yin

math.DGarXiv:1606.07877
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Abstract

We prove that the Ricci flow that contracts a hyperbolic cusp has curvature decay like one over time squared. In order to do this, we prove a new Li-Yau type differential Harnack inequality for Ricci flow on surfaces.

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Submitted
Jun 25, 2016
Subjects
math.DG, math.AP
Ricci flowHarnack inequalitiesSurfaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds (Songzi Li and Xiang-Dong Li, 2017)6Li 2017 New differential Harnack inequalities for nonlinear heat equations (Jia-Yong Wu, 2018)1Wu 2018 Parabolic frequency monotonicity on compact manifolds (Xiaolong Li and Kui Wang, 2018)7Li 2018 Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds (Cyrille Combete, Serge Degla and Leonard Todjihounde, 2018)5Combete 2018 Smoothing a measure on a Riemann surface using Ricci flow (Peter M. Topping and Hao Yin, 2021)3Topping 2021 Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces (Peter M. Topping and Hao Yin, 2023)2Topping 2023 The Harnack inequality without convexity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2026)4Sobnack 2026 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)8Deruelle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Topics Harnack inequalities, Surfaces · Shared terms “differential Harnack”

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

    Peter M. Topping and Hao Yin

    Topics Surfaces · Same authors Peter M. Topping, Hao Yin

  3. Smoothing a measure on a Riemann surface using Ricci flow

    Peter M. Topping and Hao Yin

    Topics Surfaces · Same authors Peter M. Topping, Hao Yin

  4. The Harnack inequality without convexity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Harnack inequalities · Same author Peter M. Topping

  5. Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds

    Cyrille Combete, Serge Degla and Leonard Todjihounde

    Topics Harnack inequalities · Shared terms “Yau type”

  6. Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifolds

    Songzi Li and Xiang-Dong Li

    Topics Harnack inequalities · Shared terms “differential Harnack”

  7. Parabolic frequency monotonicity on compact manifolds

    Xiaolong Li and Kui Wang

    Topics Harnack inequalities, Surfaces

  8. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al.

    Topics Harnack inequalities · Same author Peter M. Topping

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) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) 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. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  3. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.