Paper · Oct 7, 2021

Uniqueness and stability of singular Ricci flows in higher dimensions

Robert Haslhofer

math.DGarXiv:2110.03412v2
PDF Abstract page

As on arXiv

Abstract

In this short note, we observe that the Bamler-Kleiner proof of uniqueness and stability for 3-dimensional Ricci flow through singularities generalizes to singular Ricci flows in higher dimensions that satisfy an analogous canonical neighborhood property. In particular, this gives a canonical evolution through singularities for manifolds with positive isotropic curvature. The new ingredients we use are the recent classification of higher dimensional -solutions by Brendle, Daskalopoulos, Naff and Sesum, and the maximum principle for the linearized Ricci-DeTurck flow on locally conformally flat manifolds due to Chen and Wu.

Ricci flowSingularities3-manifoldsSphere theoremsWeak & 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 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 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)4Kleiner 2014 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)6Choi 2021 A note on blowup limits in 3d Ricci flow (Beomjun Choi and Robert Haslhofer, 2021)2Choi 2021 Classification of compact manifolds with positive isotropic curvature (Hong Huang, 2023)7Huang 2023 On κ-solutions and canonical neighborhoods in 4d Ricci flow (Robert Haslhofer, 2023)1Haslhofer 2023 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)5Lee 2024 On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions (Zhengnan Chen, 2025)8Chen 2025 Mean curvature flow through singularities (Robert Haslhofer, 2025)3Haslhofer 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On -solutions and canonical neighborhoods in 4d Ricci flow

    Robert Haslhofer PDF

    Topics 3-manifolds, Weak & singular flows, Singularities · Same author Robert Haslhofer · Shared terms “canonical neighborhood”

  2. A note on blowup limits in 3d Ricci flow

    Beomjun Choi and Robert Haslhofer PDF

    Topics 3-manifolds, Weak & singular flows, Singularities · Same author Robert Haslhofer

  3. Mean curvature flow through singularities

    Robert HaslhoferWider flows PDF

    Topics Weak & singular flows, Singularities · Same author Robert Haslhofer · Shared terms “higher dimensions”

  4. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics 3-manifolds, Weak & singular flows

  5. 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, Weak & singular flows, Singularities

  6. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer PDF

    Topics Weak & singular flows, Singularities · Same author Robert Haslhofer · Shared terms “Kleiner”, “Bamler”

  7. Classification of compact manifolds with positive isotropic curvature

    Hong Huang PDF

    Topics Sphere theorems, Stability & uniqueness · Shared terms “positive isotropic”, “isotropic curvature”, “Chen”

  8. On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

    Zhengnan Chen PDF

    Topics Sphere theorems · Shared terms “higher dimensions”, “positive isotropic”, “isotropic curvature”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 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.

  3. 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.