Paper · Jan 26, 2018

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

Max Hallgren

math.DGarXiv:1801.08643
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Abstract

In this short paper, we show there do not exist three-dimensional noncompact -solutions of Ricci flow that have positive curvature and satisfy a Type-I bound. This represents progress towards the proof of Perelman's conjecture that the only complete noncompact three-dimensional -solution with positive curvature is the Bryant soliton.

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Submitted
Jan 26, 2018
Subjects
math.DG

Comments: 4 pages

Ricci flowAncient solutionsSingularities3-manifoldsBoundaries & 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 three-dimensional standard solution to the Ricci flow is modeled by the Bryant soliton (Yongjia Zhang, 2017)5Zhang 2017 On three-dimensional Type I κ-solutions to the Ricci flow (Yongjia Zhang, 2017)3Zhang 2017 On the κ-solutions of the Ricci flow on noncompact 3-manifolds (Liang Cheng and Anqiang Zhu, 2018)4Cheng 2018 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)1Brendle 2018 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)8Brendle 2020 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)2Cheng 2021 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)6Lai 2022 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)7Lynch 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Ancient solutions, 3-manifolds, Boundaries & non-compact · Shared terms “Bryant soliton”, “Perelman”

  2. On the noncollapsedness of positively curved Type I ancient Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, 3-manifolds, Boundaries & non-compact · Shared terms “noncompact Type”, “dimensional noncompact”

  3. On three-dimensional Type I -solutions to the Ricci flow

    Yongjia Zhang

    Topics Ancient solutions, 3-manifolds, Boundaries & non-compact · Shared terms “dimensional noncompact”, “Bryant soliton”, “Perelman”

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

    Liang Cheng and Anqiang Zhu

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

  5. The three-dimensional standard solution to the Ricci flow is modeled by the Bryant soliton

    Yongjia Zhang

    Topics Ancient solutions, 3-manifolds · Shared terms “Bryant soliton”

  6. O(2)-symmetry of 3D steady gradient Ricci solitons

    Yi Lai

    Topics Ancient solutions, 3-manifolds · Shared terms “positive curvature”, “Bryant soliton”

  7. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego

    Topics Ancient solutions, Boundaries & non-compact, Singularities · Shared terms “progress towards”

  8. Uniqueness of compact ancient solutions to three-dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

    Topics Ancient solutions, 3-manifolds, Singularities · Shared terms “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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  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.