Paper · Nov 22, 2019

Ricci Flow Singularity for Triaxial Bianchi IX Metric

M. Syafiq Johar

math.DGarXiv:1911.09797
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Abstract

In this work, we are going to find sufficient conditions on the initial triaxial Bianchi IX metric on some 4-dimensional manifolds foliated by homogeneous S3 for a Type I singularity to occur when it is flowed under the Ricci flow. This work generalises the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as the work of Isenberg, Knopf, and Sesum, in which they introduced some ansatz for the problem setup.

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Submitted
Nov 22, 2019
Subjects
math.DG, math.AP

Comments: 26 pages, 6 figures

Ricci flowSingularities4-manifoldsHomogeneous spacesRotational 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 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 Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature (Renato G. Bettiol and Anusha M. Krishnan, 2016)2Bettiol 2016 Ricci flow of warped Berger metrics on ℝ⁴ (Francesco Di Giovanni, 2019)3Di Giovanni 2019 Exponential Asymptotics of Ricci Flow Neckpinch (Hamidreza Mahmoudian, 2023)1Mahmoudian 2023 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)4Isenberg 2025 Expanding Ricci solitons and Higgs bundles (Ramiro A. Lafuente and Adam Thompson, 2025)6Lafuente 2025 Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group (Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim, 2025)7Calvaruso 2025 Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups (Youssef Ayad, 2026)5Ayad 2026 An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow (Hongxin Guo and Guozhi Lin, 2026)8Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Exponential Asymptotics of Ricci Flow Neckpinch

    Hamidreza Mahmoudian

    Topics Rotational symmetry & warped products, Singularities · Shared terms “Isenberg”, “Knopf”, “Angenent”

  2. Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

    Renato G. Bettiol and Anusha M. Krishnan

    Topics 4-manifolds, Homogeneous spaces

  3. Ricci flow of warped Berger metrics on

    Francesco Di Giovanni

    Topics Rotational symmetry & warped products, Homogeneous spaces, Singularities · Shared terms “rotationally symmetric”

  4. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al.

    Topics 4-manifolds, Rotational symmetry & warped products, Singularities

  5. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces

  6. Expanding Ricci solitons and Higgs bundles

    Ramiro A. Lafuente and Adam Thompson

    Topics 4-manifolds, Homogeneous spaces

  7. Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

    Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim

    Topics 4-manifolds, Homogeneous spaces · Shared terms “Bianchi”

  8. An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

    Hongxin Guo and Guozhi Lin

    Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”, “Knopf”, “Angenent”

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 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.