Paper · Mar 11, 2024

Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

David Garfinkle, James Isenberg, Dan Knopf, Haotian Wu

math.DGarXiv:2403.06427
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Abstract

Kröncke has shown that the Fubini-Study metric is an unstable generalized stationary solution of Ricci flow [Krö20]. In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini-Study metric develop local singularities modeled by the blowdown soliton discovered in [FIK03].

Comments: 14 pages, 3 figures. Comments are welcome!

Ricci flowSingularitiesStability & uniquenessNumerics & visualization

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows PDF

    Topics Numerics & visualization, Singularities, Stability & uniqueness · Same authors David Garfinkle, James Isenberg, Dan Knopf · Shared terms “numerical simulations”, “perturbation”

  2. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al. PDF

    Topics Singularities · Same authors James Isenberg, Dan Knopf · Shared terms “local singularity”

  3. Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

    Sigurd B. Angenent and Dan Knopf PDF

    Topics Stability & uniqueness · Same author Dan Knopf · Shared terms “unstable perturbations”, “stationary”

  4. Stability of piecewise flat Ricci flow

    Rory Conboye PDF

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical simulations”

  5. Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes

    Davide De Biasio, Julian Freigang, Dieter Lust et al. PDF

    Topics Numerics & visualization, Singularities, Stability & uniqueness · Shared terms “unstable”, “perturbation”

  6. Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow

    Jihui Wang, Matteo Lulli and Antonino Marciano PDF

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “perturbation”, “unstable”, “simulations”

  7. Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

    Roman BahadursinghWider flows PDF

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical simulations”, “perturbation”

  8. Which shapes can appear in a Curve Shortening Flow Singularity?

    Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al.Wider flows PDF

    Topics Numerics & visualization, Singularities · Shared terms “numerical simulations”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  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.