Paper · Dec 10, 2018

Ancient solutions to the Ricci flow in higher dimensions

Xiaolong Li, Yongjia Zhang

math.DGarXiv:1812.04156v2
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Abstract

In this paper, we study -noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder . By making use of the properties of such ancient solutions, we generalize part one of Brendle to higher dimensions, that is, every noncompact -noncollapsed rotationally symmetric ancient solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton.

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Submitted
Dec 10, 2018
Last revised
May 14, 2020 (v2)
Subjects
math.DG, math.AP

Comments: Final version, to appear in Communications in Analysis and Geometry

Ricci flowRicci solitonsAncient solutions(Non)collapsingSphere theoremsBoundaries & non-compactRotational 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 Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)2Zhang 2017 On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)5Zhang 2017 Uniqueness of convex ancient solutions to mean curvature flow in ℝ³ (S. Brendle and K. Choi, 2017)6Brendle 2017 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)4Brendle 2018 Rotational symmetry of ancient solutions to the Ricci flow in dimension 3 – The compact case (S. Brendle, 2019)8Brendle 2019 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)1Brendle 2020 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)3Cheng 2021 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. Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

    Simon Brendle and Keaton Naff

    Topics Sphere theorems, Rotational symmetry & warped products, (Non)collapsing · Shared terms “higher dimensions”, “Bryant soliton”, “noncollapsed”

  2. Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

    Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions · Same author Yongjia Zhang · Shared terms “curvature operator”, “nonnegative curvature”

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

    Liang Cheng and Yongjia Zhang

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Same author Yongjia Zhang · Shared terms “noncollapsed”

  4. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    S. Brendle and K. ChoiWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Shared terms “higher dimensions”, “noncollapsed”, “rotationally symmetric”

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

    Yongjia Zhang

    Topics (Non)collapsing, Ancient solutions · Same author Yongjia Zhang · Shared terms “curvature operator”, “nonnegative curvature”

  6. Uniqueness of convex ancient solutions to mean curvature flow in

    S. Brendle and K. Choi

    Topics Sphere theorems, Rotational symmetry & warped products, (Non)collapsing · Shared terms “noncollapsed”, “rotationally symmetric”

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

    Stephen Lynch and Andoni Royo Abrego

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Shared terms “higher dimensions”, “noncollapsed”

  8. Rotational symmetry of ancient solutions to the Ricci flow in dimension – The compact case

    S. Brendle

    Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Shared terms “Bryant soliton”, “rotationally symmetric”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 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.

  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.