Paper · Apr 7, 2021

Ricci Flow with Ricci Curvature and Volume Bounded Below

Max Hallgren

math.DGarXiv:2104.03386v2
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Abstract

We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a orbifold at any finite-time singularity, so has an extension through the singularity via orbifold Ricci flow. Moreover, a Type-I blowup of the flow based at any orbifold point converges to a flat cone in the Gromov-Hausdorff sense, without passing to a subsequence. In addition, we prove bounds for the curvature tensor on time-slices for any . In higher dimensions, we show that every singular point of the flow is a Type-II point, and that any tangent flow at a singular point is a static flow corresponding to a Ricci flat cone.

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Submitted
Apr 7, 2021
Last revised
Feb 28, 2022 (v2)
Subjects
math.DG

Comments: Corrected several typos, improved exposition in Section 5

Ricci flowRicci-flat & Calabi–YauSingularities4-manifoldsLimit spaces & RCD

Its neighbourhood

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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 The branching curves and their application to the four dimensional Ricci flow (Ilias Tergiakidis, 2018)3Tergiakidis 2018 The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds (Max Hallgren, 2020)2Hallgren 2020 Ricci flow with bounded curvature integrals (Shota Hamanaka, 2021)5Hamanaka 2021 Harmonic spinors in the Ricci flow (Julius Baldauf, 2022)8Baldauf 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)1Hallgren 2023 Remarks on Singular Kähler-Einstein Metrics (Max Hallgren and Gábor Székelyhidi, 2025)4Hallgren 2025 Singular sets in noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)7Fang 2025 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)6Cifarelli 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Limit spaces & RCD, Singularities · Same author Max Hallgren · Shared terms “flow based”, “time slice”, “Gromov Hausdorff”

  2. The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds

    Max Hallgren

    Topics 4-manifolds, Singularities · Same author Max Hallgren · Shared terms “orbifold”, “time singularity”, “finite time”

  3. The branching curves and their application to the four dimensional Ricci flow

    Ilias Tergiakidis

    Topics 4-manifolds, Limit spaces & RCD, Singularities · Shared terms “dimensional Ricci”, “four dimensional”, “Gromov”

  4. Remarks on Singular Kähler-Einstein Metrics

    Max Hallgren and Gábor Székelyhidi

    Topics Ricci-flat & Calabi–Yau, Limit spaces & RCD, Singularities · Same author Max Hallgren

  5. Ricci flow with bounded curvature integrals

    Shota Hamanaka

    Topics 4-manifolds, Singularities · Shared terms “orbifold”, “higher dimensions”, “finite time”

  6. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al.

    Topics Singularities · Same author Max Hallgren · Shared terms “time singularity”, “finite time”, “tangent flow”

  7. Singular sets in noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li

    Topics 4-manifolds, Limit spaces & RCD, Singularities · Shared terms “tangent flow”, “uniformly bounded”, “four dimensional”

  8. Harmonic spinors in the Ricci flow

    Julius Baldauf

    Topics Ricci-flat & Calabi–Yau, 4-manifolds · Shared terms “simply connected”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.