Paper · Dec 27, 2019

The full holonomy group under the Ricci flow

Mary Cook, Brett Kotschwar

math.DGarXiv:1912.12240v2
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Abstract

We give a short, direct proof that the full holonomy group of a solution to the Ricci flow is invariant up to isomorphism using the invariance of the reduced holonomy under the flow.

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Submitted
Dec 27, 2019
Last revised
Sep 4, 2020 (v2)
Subjects
math.DG

Comments: v2: Minor corrections. 9 pages, no figures. To appear in Proc. Amer. Math. Soc

Ricci flow

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 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 Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones (Brett Kotschwar, 2017)8Kotschwar 2017 Isometries of asymptotically conical shrinking Ricci solitons (Brett Kotschwar and Lu Wang, 2018)7Kotschwar 2018 On the maximal rate of convergence under the Ricci flow (Brett Kotschwar, 2020)2Kotschwar 2020 Preservation of product structures under the Ricci flow with instantaneous curvature bounds (Mary Cook, 2022)4Cook 2022 Ricci flows which terminate in cones (Brett Kotschwar, 2023)3Kotschwar 2023 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)6Kotschwar 2023 Cohomogeneity one solutions of the IIB system (Lorenzo Foscolo and Mario Garcia-Fernandez, 2026)5Foscolo 2026 Chern holonomy under the first and second Chern–Ricci flows (Yuan Liao, 2026)1Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Chern holonomy under the first and second Chern–Ricci flows

    Yuan Liao

    Topics Ricci flow · Shared terms “holonomy”

  2. On the maximal rate of convergence under the Ricci flow

    Brett Kotschwar

    Topics Ricci flow · Same author Brett Kotschwar

  3. Ricci flows which terminate in cones

    Brett Kotschwar

    Topics Ricci flow · Same author Brett Kotschwar

  4. Preservation of product structures under the Ricci flow with instantaneous curvature bounds

    Mary Cook

    Topics Ricci flow · Same author Mary Cook

  5. Cohomogeneity one solutions of the IIB system

    Lorenzo Foscolo and Mario Garcia-Fernandez

    Topics Ricci flow · Shared terms “full holonomy”

  6. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Ricci flow · Same author Brett Kotschwar

  7. Isometries of asymptotically conical shrinking Ricci solitons

    Brett Kotschwar and Lu Wang

    Same author Brett Kotschwar

  8. Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones

    Brett Kotschwar

    Same author Brett Kotschwar

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)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.