Paper · Oct 21, 2018

On the solutions of the Ricci flow on noncompact 3-manifolds

Liang Cheng, Anqiang Zhu

math.DGarXiv:1810.08934
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper we prove that there is no -solution of Ricci flow on 3-dimensional noncompact manifold with strictly positive sectional curvature and blow up at some finite time satisfying for some point . This partially confirms a conjecture of Perelman.

View PDF View HTML arXiv page
Submitted
Oct 21, 2018
Subjects
math.DG
Ricci flowSingularities3-manifoldsSphere theoremsBoundaries & non-compact

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 The Nonexistence of Noncompact Type-I Ancient 3-d κ-Solutions of Ricci Flow with Positive Curvature (Max Hallgren, 2018)4Hallgren 2018 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)8Brendle 2018 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)1Cheng 2021 A no expanding breather theorem for noncompact Ricci flows (Liang Cheng and Yongjia Zhang, 2021)3Cheng 2021 Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds (Liang Cheng and Yongjia Zhang, 2021)2Cheng 2021 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)6Lee 2022 On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces (Nurlan Abiev, 2023)5Abiev 2023 Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen (David González-Álvaro and Masoumeh Zarei, 2024)7González-Álv 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the noncollapsedness of positively curved Type I ancient Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Sphere theorems, 3-manifolds, Boundaries & non-compact · Same author Liang Cheng · Shared terms “dimensional noncompact”, “positive sectional”, “sectional curvature”

  2. Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds

    Liang Cheng and Yongjia Zhang

    Topics Boundaries & non-compact · Same author Liang Cheng · Shared terms “noncompact manifolds”, “partially”, “sectional curvature”

  3. A no expanding breather theorem for noncompact Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Sphere theorems, Boundaries & non-compact · Same author Liang Cheng

  4. The Nonexistence of Noncompact Type-I Ancient 3-d -Solutions of Ricci Flow with Positive Curvature

    Max Hallgren

    Topics 3-manifolds, Boundaries & non-compact, Singularities · Shared terms “dimensional noncompact”, “Perelman”

  5. On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces

    Nurlan Abiev

    Topics Sphere theorems, 3-manifolds · Shared terms “positive sectional”, “sectional curvature”

  6. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping

    Topics Sphere theorems, 3-manifolds, Boundaries & non-compact · Shared terms “sectional curvature”

  7. Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen

    David González-Álvaro and Masoumeh Zarei

    Topics Sphere theorems · Shared terms “positive sectional”, “sectional curvature”

  8. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics 3-manifolds, Boundaries & non-compact, Singularities · Shared terms “finite time”, “Perelman”

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 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  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.