Paper · Mar 15, 2018

Blowup criteria for geometric flows on surfaces

Lothar Schiemanowski

math.DGarXiv:1803.05737
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Abstract

Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics on surfaces with a uniform bound on their volumes, square integrals of their curvatures and injectivity radii. In particular we prove blow up criteria for the harmonic Ricci flow and for the spinor flow on surfaces.

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Submitted
Mar 15, 2018
Subjects
math.DG

Comments: 23 pages

Ricci flowSingularitiesSurfacesLimit spaces & RCD

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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 Analysis of Type I Singularities in the Harmonic Ricci Flow (Gianmichele Di Matteo, 2018)3Di Matteo 2018 Convergence of Hermitian manifolds and the Type IIB flow (Nikita Klemyatin, 2021)4Klemyatin 2021 Ricci flow on surfaces along the standard lightcone in the 3+1-Minkowski spacetime (Markus Wolff, 2022)8Wolff 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)7Hallgren 2023 Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry (Georgios Papadopoulos and Edward Witten, 2024)6Papadopoulos 2024 Contractive transport maps from S² to nearly spherical surfaces with positive Ricci curvature (Jordan Serres, 2025)2Serres 2025 Path Connectivity of Anosov Metrics on Surfaces (Guilherme Brandão Guglielmo and R. Ruggiero, 2026)5Guglielmo 2026 Notes on harmonic-Ricci flow on surface (Xiang-Zhi Cao, 2026)1Cao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao

    Topics Surfaces · Shared terms “harmonic Ricci”

  2. Contractive transport maps from to nearly spherical surfaces with positive Ricci curvature

    Jordan Serres

    Topics Surfaces, Limit spaces & RCD

  3. Analysis of Type I Singularities in the Harmonic Ricci Flow

    Gianmichele Di Matteo

    Topics Limit spaces & RCD, Singularities · Shared terms “harmonic Ricci”, “compactness theorem”

  4. Convergence of Hermitian manifolds and the Type IIB flow

    Nikita Klemyatin

    Topics Limit spaces & RCD, Singularities · Shared terms “criteria”, “compactness theorem”

  5. Path Connectivity of Anosov Metrics on Surfaces

    Guilherme Brandão Guglielmo and R. Ruggiero

    Topics Surfaces · Shared terms “geometric flows”

  6. Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry

    Georgios Papadopoulos and Edward Witten

    Topics Surfaces, Limit spaces & RCD · Shared terms “compactness”

  7. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Limit spaces & RCD, Singularities

  8. Ricci flow on surfaces along the standard lightcone in the -Minkowski spacetime

    Markus Wolff

    Topics Surfaces, Singularities

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.