Paper · May 4, 2018

A note on existence of exhaustion functions and its applications

Shaochuang Huang

math.DGarXiv:1805.01840
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Abstract

In this note, we prove an existence result on exhaustion functions adapting the method by L.-F. Tam. Then we apply it to prove short-time existence of Ricci flow and study Yau's uniformization conjecture using similar method as Fei He and Lee-Tam.

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Submitted
May 4, 2018
Subjects
math.DG

Comments: 11 pages

Ricci flowSurfacesHarmonic 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 Uniqueness of Weak Solutions for the Normalised Ricci Flow in Two Dimensions (Franziska Borer, 2016)7Borer 2016 A simple proof of the short-time existence and uniqueness for Ricci flow (Kaveh Eftekharinasab, 2016)2Eftekharinas 2016 On existence and curvature estimates of Ricci flow (Man-Chun Lee and Luen-Fai Tam, 2017)6Lee 2017 Short-Time Existence Theorem for the CR Torsion Flow (Shu-Cheng Chang and Chin-Tung Wu, 2018)4Chang 2018 Ricci flow on Finsler surfaces (Behroz Bidabad and Maral K. Sedaghat, 2018)8Bidabad 2018 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)1Huang 2021 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)5Bidabad 2023 Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds (Yuang Shi, 2025)3Shi 2025
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 for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick · Same author Shaochuang Huang · Shared terms “short time”, “Tam”, “time existence”

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

    Kaveh Eftekharinasab

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

  3. Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds

    Yuang Shi

    Topics Surfaces · Shared terms “Yau uniformization”, “Lee Tam”, “uniformization conjecture”

  4. Short-Time Existence Theorem for the CR Torsion Flow

    Shu-Cheng Chang and Chin-Tung Wu

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

  5. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

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

  6. On existence and curvature estimates of Ricci flow

    Man-Chun Lee and Luen-Fai Tam

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

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

    Franziska Borer

    Topics Harmonic maps & DeTurck trick, Surfaces

  8. Ricci flow on Finsler surfaces

    Behroz Bidabad and Maral K. Sedaghat

    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) 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) 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. 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.