Paper · Mar 8, 2017

Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons

James Isenberg, Dan Knopf, Natasa Sesum

math.DGarXiv:1703.02918v5
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Abstract

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these solutions is expected to be the "blowdown soliton" discovered in [FIK03]. Our partial results support the conjecture that the blowdown soliton is stable under Ricci flow, as well as the conjectured stability of the subspace of Kahler metrics under Ricci flow.

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Submitted
Mar 8, 2017
Last revised
Mar 5, 2019 (v5)
Subjects
math.DG

Comments: Revised to indicate dependence on two technical conjectures for which we present formal arguments

Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsSingularitiesKähler geometryStability & 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 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 Dynamic instability of ℂP}^N under Ricci flow (Dan Knopf and Natasa Sesum, 2017)3Knopf 2017 Kähler-Ricci flow on horospherical manifold (François Delgove, 2019)6Delgove 2019 Singularity formation of complete Ricci flow solutions (Timothy Carson, James Isenberg, Dan Knopf et al., 2020)1Carson 2020 On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)5Cifarelli 2022 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., 2022)8Garfinkle 2022 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)2Garfinkle 2024 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)4Isenberg 2025 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)7Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularity formation of complete Ricci flow solutions

    Timothy Carson, James Isenberg, Dan Knopf et al.

    Topics Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf, Natasa Sesum · Shared terms “conjectured stability”, “singularity models”, “evidence”

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

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf · Shared terms “blowdown soliton”, “singularity modeled”, “soliton discovered”

  3. Dynamic instability of under Ricci flow

    Dan Knopf and Natasa Sesum

    Topics Kähler geometry, Stability & uniqueness · Same authors Dan Knopf, Natasa Sesum · Shared terms “expected”, “subspace”

  4. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al.

    Topics Singularities, Ricci solitons · Same authors James Isenberg, Dan Knopf, Natasa Sesum · Shared terms “singularity models”

  5. On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Shared terms “time Type”, “finite time”

  6. Kähler-Ricci flow on horospherical manifold

    François Delgove

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry

  7. Finite Time Type I Singularities of the Kähler Ricci Flow

    Tongxin Xu and Zhenlei Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time Type”, “finite time”

  8. 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

    Topics Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf · Shared terms “evidence”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  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.