Paper · Nov 21, 2023

Exponential Asymptotics of Ricci Flow Neckpinch

Hamidreza Mahmoudian

math.DGarXiv:2311.12725
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Abstract

In this article we examine the formation of cylindrical neckpinch singularities in Ricci flow in compact manifolds. Rigorous examples of neckpinch sigularity for rotationally symmetric initial data were first constructed by Angenent and Knopf, and examples with given asymptotic behaviour were constructed by them and Isenberg. Here we discuss the asymptotic behavior of the flow under Type-I assumption for general symmetric initial data, and show the previously constructed asymptotic profiles and are the only possibilities.

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Submitted
Nov 21, 2023
Subjects
math.DG
Ricci flowSingularitiesRotational 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 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 Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions (Kin Ming Hui, 2021)1Hui 2021 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)6Garfinkle 2022 Symmetries of Ricci Flows (Enrique López, Stylianos Dimas and Yuri Bozhkov, 2022)8López 2022 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)7Fujihara 2024 A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary (Gang Li, 2026)4Li 2026 A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II (Gang Li, 2026)5Li 2026 Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers (Peng Peng, 2026)3Peng 2026 An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow (Hongxin Guo and Guozhi Lin, 2026)2Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

    Kin Ming Hui

    Topics Rotational symmetry & warped products · Shared terms “asymptotic behaviour”, “rotationally symmetric”

  2. An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

    Hongxin Guo and Guozhi Lin

    Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”, “Knopf”, “Angenent”

  3. Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

    Peng PengWider flows

    Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”, “Angenent”

  4. A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary

    Gang Li

    Topics Rotational symmetry & warped products · Shared terms “compact manifolds”, “rotationally symmetric”

  5. A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II

    Gang Li

    Topics Rotational symmetry & warped products · Shared terms “compact manifolds”, “rotationally symmetric”

  6. 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 Rotational symmetry & warped products, Singularities · Shared terms “neckpinch”, “rotationally symmetric”

  7. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows

    Topics Rotational symmetry & warped products, Singularities

  8. Symmetries of Ricci Flows

    Enrique López, Stylianos Dimas and Yuri Bozhkov

    Topics Rotational symmetry & warped products

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.