Paper · Feb 12, 2025

Local singularities of compact multiply warped Ricci flow solutions

James Isenberg, Dan Knopf, Zilu Ma, Natasa Sesum

math.DGarXiv:2502.08500
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Abstract

We demonstrate that any four-dimensional shrinking Ricci soliton , where is any two-dimensional complete noncompact surface and is a warped product metric over the base , has to be isometric to the generalized cylinder equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products – but not products – and provide rigorous examples of the formation of generalized cylinder singularity models .

Ricci flowRicci solitonsSingularities4-manifoldsSurfacesBoundaries & non-compactRotational symmetry & warped products

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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 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 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)3Garfinkle 2022 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)7Bahuaud 2023 Dimension Reduction for Positively Curved Steady Solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)8Chan 2023 Unique Asymptotics of Steady Ricci Solitons with Symmetry (Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum, 2023)2Ma 2023 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)4Kotschwar 2023 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)1Garfinkle 2024 Asymptotic Geometry of Four-Dimensional Steady Solitons (Aprameya Girish Hebbar and Natasa Sesum, 2026)5Hebbar 2026 On Ricci solitons whose level hypersurfaces have parallel second fundamental form (Matheus Andrade Ribeiro de Moura Horácio, 2026)6Horácio 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

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

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

  2. Unique Asymptotics of Steady Ricci Solitons with Symmetry

    Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum PDF

    Topics 4-manifolds, Ricci solitons · Same authors Zilu Ma, Natasa Sesum

  3. 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 Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same authors James Isenberg, Dan Knopf

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

    Brett Kotschwar PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Ricci solitons · Shared terms “multiply warped”

  5. Asymptotic Geometry of Four-Dimensional Steady Solitons

    Aprameya Girish Hebbar and Natasa Sesum PDF

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Same author Natasa Sesum · Shared terms “four dimensional”, “complete noncompact”

  6. On Ricci solitons whose level hypersurfaces have parallel second fundamental form

    Matheus Andrade Ribeiro de Moura Horácio PDF

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “multiply warped”, “shrinking Ricci”

  7. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al. PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same author James Isenberg

  8. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics 4-manifolds, Singularities, Ricci solitons · Same author Zilu Ma

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) 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) 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)

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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.