Paper · Apr 20, 2016

A simple proof of the short-time existence and uniqueness for Ricci flow

Kaveh Eftekharinasab

math.DGarXiv:1604.05853v3
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Abstract

In this short note, we give simple proof of the Ricci flow's local existence and uniqueness on closed Einstein manifolds. We suggest a new setting for studying the space of Riemannian metrics on a compact manifold.

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Submitted
Apr 20, 2016
Last revised
Nov 8, 2022 (v3)
Subjects
math.DG
Journal
Comptes rendus de lAcademie bulgare des Sciences, vol. 72, no. 5, 2019, 569-572
DOI
10.7546/CRABS.2019.05.01

Comments: arXiv admin note: text overlap with arXiv:1508.05239

Ricci flowEinstein metricsStability & 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 Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below (Raphael Hochard, 2016)5Hochard 2016 Short-Time Existence Theorem for the CR Torsion Flow (Shu-Cheng Chang and Chin-Tung Wu, 2018)1Chang 2018 Ricci flow on Finsler surfaces (Behroz Bidabad and Maral K. Sedaghat, 2018)4Bidabad 2018 Second Ricci flow on noncompact Hermitian manifolds (Man-Chun Lee, 2018)7Lee 2018 Short time existence of the classical solution to the fractional Mean curvature flow (Vesa Julin and Domenico La Manna, 2019)8Julin 2019 Finslerian geodesics on Fréchet manifolds (Kaveh Eftekharinasab and Valentyna Petrusenko, 2020)6Eftekharinas 2020 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)3Huang 2021 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)2Bidabad 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Short-Time Existence Theorem for the CR Torsion Flow

    Shu-Cheng Chang and Chin-Tung Wu

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”, “local existence”

  2. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  3. 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 “short time”, “time existence”

  4. Ricci flow on Finsler surfaces

    Behroz Bidabad and Maral K. Sedaghat

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  5. Short-time existence of the Ricci flow on complete, non-collapsed -manifolds with Ricci curvature bounded from below

    Raphael Hochard

    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”

  6. Finslerian geodesics on Fréchet manifolds

    Kaveh Eftekharinasab and Valentyna Petrusenko

    Topics Einstein metrics · Same author Kaveh Eftekharinasab · Shared terms “Riemannian metrics”

  7. Second Ricci flow on noncompact Hermitian manifolds

    Man-Chun Lee

    Topics Harmonic maps & DeTurck trick, Einstein metrics · Shared terms “short time”, “time existence”

  8. Short time existence of the classical solution to the fractional Mean curvature flow

    Vesa Julin and Domenico La MannaWider flows

    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.