Paper · Apr 30, 2019

Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

Maxwell Stolarski

math.DGarXiv:1905.00087
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Abstract

For any manifold admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any there exists a solution with curvature blow-up rate with singularity modeled on a Ricci-flat cone at parabolic scales.

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Submitted
Apr 30, 2019
Subjects
math.DG

Comments: 134 pages. Comments welcome

Ricci flowEinstein metricsRicci-flat & Calabi–YauSingularitiesRotational symmetry & warped products

Its neighbourhood

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 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 A Pohozaev Identity on Warped Product Solitons (Timothy Carson, 2016)2Carson 2016 Ricci Flow recovering from pinched discs (Timothy Carson, 2017)1Carson 2017 Ricci Almost Solitons on semi-Riemannian Warped Products (Keti Tenenblat and Valter Borges, 2017)5Tenenblat 2017 Ricci flow from some spaces with asymptotically cylindrical singularities (Timothy Carson, 2018)3Carson 2018 Characteristics of conformal Ricci soliton on warped product spaces (Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya, 2021)8Ganguly 2021 Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers (Maxwell Stolarski, 2022)7Stolarski 2022 Classification of Gradient Ricci solitons with harmonic Weyl curvature (Jongsu Kim, 2023)6Kim 2023 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)4Kotschwar 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flow recovering from pinched discs

    Timothy Carson

    Topics Rotational symmetry & warped products, Singularities · Shared terms “doubly warped”, “product metrics”, “singularity modeled”

  2. A Pohozaev Identity on Warped Product Solitons

    Timothy Carson

    Topics Rotational symmetry & warped products, Singularities · Shared terms “product metrics”

  3. Ricci flow from some spaces with asymptotically cylindrical singularities

    Timothy Carson

    Topics Rotational symmetry & warped products, Singularities · Shared terms “doubly warped”

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

    Brett Kotschwar

    Topics Rotational symmetry & warped products, Einstein metrics

  5. Ricci Almost Solitons on semi-Riemannian Warped Products

    Keti Tenenblat and Valter Borges

    Topics Ricci-flat & Calabi–Yau, Rotational symmetry & warped products, Einstein metrics

  6. Classification of Gradient Ricci solitons with harmonic Weyl curvature

    Jongsu Kim

    Topics Ricci-flat & Calabi–Yau, Rotational symmetry & warped products, Einstein metrics

  7. Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

    Maxwell Stolarski

    Topics Singularities · Same author Maxwell Stolarski · Shared terms “singularity modeled”, “singularity formation”

  8. Characteristics of conformal Ricci soliton on warped product spaces

    Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya

    Topics Ricci-flat & Calabi–Yau, Rotational symmetry & warped products · Shared terms “admitting”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 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.

  2. 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.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.