Paper · Jan 26, 2016

Uniqueness of Weak Solutions for the Normalised Ricci Flow in Two Dimensions

Franziska Borer

math.AParXiv:1601.06853
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As on arXiv

Abstract

We show uniqueness of classical solutions of the normalised two-dimensional Hamilton-Ricci flow on closed, smooth manifolds for smooth data among solutions satisfying (essentially) only a uniform bound for the Liouville energy and a natural space-time -bound for the time derivative of the solution. The result is surprising when compared with results for the harmonic map heat flow, where non-uniqueness through reverse bubbling may occur.

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Submitted
Jan 26, 2016
Subjects
math.AP, math.DG
Ricci flowSurfacesWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trick

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 On the stability of harmonic maps under the homogeneous Ricci flow (Rafaela F. do Prado and Lino Grama, 2017)4Prado 2017 A note on existence of exhaustion functions and its applications (Shaochuang Huang, 2018)7Huang 2018 On the regularity of Ricci flows coming out of metric spaces (Alix Deruelle, Felix Schulze and Miles Simon, 2019)5Deruelle 2019 On the nonexistence of harmonic and bi-harmonic maps (Ahmed Mohammed Cherif, 2020)8Cherif 2020 Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry (Keita Kunikawa and Yohei Sakurai, 2021)3Kunikawa 2021 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)2Huang 2021 On harmonic and biharmonic maps from gradient Ricci solitons (Volker Branding, 2022)6Branding 2022 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)1Lee 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Stability & uniqueness · Shared terms “map heat”, “heat flow”

  2. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “map heat”, “heat flow”

  3. Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

    Keita Kunikawa and Yohei Sakurai

    Topics Harmonic maps & DeTurck trick, Weak & singular flows · Shared terms “map heat”, “heat flow”, “Liouville”

  4. On the stability of harmonic maps under the homogeneous Ricci flow

    Rafaela F. do Prado and Lino Grama

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness

  5. On the regularity of Ricci flows coming out of metric spaces

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Harmonic maps & DeTurck trick, Weak & singular flows · Shared terms “map heat”, “heat flow”, “Hamilton”

  6. On harmonic and biharmonic maps from gradient Ricci solitons

    Volker Branding

    Topics Harmonic maps & DeTurck trick, Surfaces

  7. A note on existence of exhaustion functions and its applications

    Shaochuang Huang

    Topics Harmonic maps & DeTurck trick, Surfaces

  8. On the nonexistence of harmonic and bi-harmonic maps

    Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.