Paper · Mar 24, 2026

The Ricci-DeTurck flow on complete manifolds

Jing-Bin Cai, Bing Wang

math.DGarXiv:2603.22834
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Abstract

Based on the framework of Koch-Lamm and tensor heat kernel estimates, we obtain a uniform proof of the short-time existence, uniqueness, and continuous dependence for Ricci flows starting from a complete Riemannian metric with bounded curvature. A new ingredient is an effective continuous dependence estimate without the assumption of injectivity radius lower bound.

Ricci flowHeat kernelStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trick

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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 2021, 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 Ricci flow of W^(2,2)-metrics in four dimensions (Tobias Lamm and Miles Simon, 2021)7Lamm 2021 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)5Huang 2021 ADM mass for C⁰ metrics and distortion under Ricci-DeTurck flow (Paula Burkhardt-Guim, 2022)3Burkhardt-Gu 2022 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)2Li 2023 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)8Bahuaud 2023 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)4Lee 2024 Well-posedness of mean curvature flow (Yongheng Han, 2026)1Han 2026 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)6Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Well-posedness of mean curvature flow

    Yongheng HanWider flows PDF

    Topics Heat kernel, Stability & uniqueness · Shared terms “continuous dependence”, “kernel estimates”, “flow starting”

  2. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang PDF

    Topics Heat kernel · Same author Bing Wang

  3. ADM mass for metrics and distortion under Ricci-DeTurck flow

    Paula Burkhardt-Guim PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “Riemannian metric”

  4. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “flow starting”, “short time”, “lower bound”

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

    Shaochuang Huang and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact, Stability & uniqueness · Shared terms “complete manifolds”, “short time”, “time existence”

  6. Heat kernel on Ricci shrinker metric measure spaces

    Bing Wang and Jie Wang PDF

    Topics Heat kernel · Same author Bing Wang

  7. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “Riemannian metric”

  8. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al. PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact, Stability & uniqueness

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) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  3. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.