Paper · Mar 8, 2023

Convergence stability for Ricci flow on manifolds with bounded geometry

Eric Bahuaud, Christine Guenther, James Isenberg, Rafe Mazzeo

math.DGarXiv:2303.04800v2
PDF Abstract page

As on arXiv

Abstract

We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on our recent work where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in Hölder norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric.

Comments: 11 pages. This revision clarifies one argument in our published article appearing in the Proceedings of the American Mathematical Society. There are small modifications to the statements of several Theorems

Ricci flowSingularitiesStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trickRotational symmetry & warped products

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 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 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)6Huang 2021 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)2Garfinkle 2022 ADM mass for C⁰ metrics and distortion under Ricci-DeTurck flow (Paula Burkhardt-Guim, 2022)8Burkhardt-Gu 2022 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)7Lee 2024 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)4Isenberg 2025 Rotationally symmetric Ricci Flow on ℝⁿ⁺¹ (Ming Hsiao, 2025)3Hsiao 2025 On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume (Ruojing Jiang and Franco Vargas Pallete, 2025)1Jiang 2025 The Ricci-DeTurck flow on complete manifolds (Jing-Bin Cai and Bing Wang, 2026)5Cai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

    Ruojing Jiang and Franco Vargas Pallete PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “hyperbolic metric”, “Hölder norm”, “sufficiently close”

  2. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same author James Isenberg · Shared terms “rotationally symmetric”, “sufficiently”

  3. Rotationally symmetric Ricci Flow on

    Ming Hsiao PDF

    Topics Harmonic maps & DeTurck trick, Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”

  4. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al. PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same author James Isenberg

  5. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang PDF

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

  6. 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 metrics”

  7. 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, Singularities, Stability & uniqueness

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

    Paula Burkhardt-Guim PDF

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

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 usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 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.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.