Paper · Oct 17, 2024

Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

Steven Gindi

math.DGarXiv:2410.13188
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Abstract

We use Lott's functional and construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature. Consequently, we prove that the blowdown limit is locally an expanding Ricci soliton when the structure group is the three dimensional Heisenberg group. In addition, we classify this soliton when the base manifold is one dimensional. This, together with Lott's work in the abelian setting, yields a complete local classification of invariant Ricci flow blowdown limits on four dimensional, nilpotent principal bundles.

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Submitted
Oct 17, 2024
Subjects
math.DG
Ricci flowRicci solitons3-manifolds4-manifoldsHomogeneous spacesStability & uniqueness

Its neighbourhood

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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 Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups (Rohollah Bakhshandeh-Chamazkoti, 2021)4Bakhshandeh- 2021 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)1Gindi 2021 Expanding Ricci solitons and Higgs bundles (Ramiro A. Lafuente and Adam Thompson, 2025)2Lafuente 2025 Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group (Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim, 2025)6Calvaruso 2025 Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups (Youssef Ayad, 2026)8Ayad 2026 Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups (Youssef Ayad, 2026)5Ayad 2026 Generalized Ricci Solitons on Heisenberg and Oscillator Algebras (Vicente Cortés, Shubham Dwivedi and Lorenz Jetter, 2026)7Cortés 2026 Ricci flows and bracket flows on twisted principal bundles (Henry Wood, 2026)3Wood 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets

    Topics 4-manifolds, Homogeneous spaces · Same author Steven Gindi · Shared terms “nilpotent symmetry”, “blowdown limits”, “four dimensional”

  2. Expanding Ricci solitons and Higgs bundles

    Ramiro A. Lafuente and Adam Thompson

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “nilpotent symmetry”, “expanding Ricci”

  3. Ricci flows and bracket flows on twisted principal bundles

    Henry Wood

    Topics Homogeneous spaces, Ricci solitons · Shared terms “principal bundles”, “invariant Ricci”, “nilpotent”

  4. Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

    Rohollah Bakhshandeh-Chamazkoti

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “nilpotent”

  5. Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “nilpotent”, “four dimensional”, “together”

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

    Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim

    Topics 4-manifolds, 3-manifolds, Homogeneous spaces · Shared terms “four dimensional”, “expanding Ricci”

  7. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

    Vicente Cortés, Shubham Dwivedi and Lorenz Jetter

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “dimensional Heisenberg”, “nilpotent”, “four dimensional”

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

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “four dimensional”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

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    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.