Paper · May 3, 2017

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

Yongjia Zhang

math.DGarXiv:1705.01248v2
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Abstract

It came to my attention after posting this paper that Yu Ding has proved the same result before. I would like to apologize to Yu Ding for the appearance of this paper.

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Submitted
May 3, 2017
Last revised
May 8, 2017 (v2)
Subjects
math.DG

Comments: Result is covered by previous work

Ricci flowAncient solutions3-manifolds

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 On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)3Zhang 2017 On three-dimensional Type I κ-solutions to the Ricci flow (Yongjia Zhang, 2017)1Zhang 2017 The Nonexistence of Noncompact Type-I Ancient 3-d κ-Solutions of Ricci Flow with Positive Curvature (Max Hallgren, 2018)7Hallgren 2018 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)2Brendle 2018 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)4Li 2018 Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons (Shun Maeta, 2019)6Maeta 2019 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)8Cheng 2021 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)5Lai 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On three-dimensional Type I -solutions to the Ricci flow

    Yongjia Zhang

    Topics Ancient solutions, 3-manifolds · Same author Yongjia Zhang · Shared terms “Bryant soliton”, “Ding”

  2. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

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

  3. On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow

    Yongjia Zhang

    Topics Ancient solutions, 3-manifolds · Same author Yongjia Zhang

  4. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Ancient solutions · Same author Yongjia Zhang · Shared terms “Bryant soliton”

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

    Yi Lai

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

  6. Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons

    Shun Maeta

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

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

    Max Hallgren

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

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

    Liang Cheng and Yongjia Zhang

    Topics Ancient solutions, 3-manifolds · Same author Yongjia Zhang

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) 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. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.