Paper · Oct 14, 2020

A family of 3d steady gradient solitons that are flying wings

Yi Lai

math.DGarXiv:2010.07272v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension , we find a family of -symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

View PDF View HTML arXiv page
Submitted
Oct 14, 2020
Last revised
Nov 29, 2020 (v2)
Subjects
math.DG

Comments: 25 pages, a figure added, minor changes

Ricci solitons(Non)collapsing3-manifoldsSphere theoremsScalar curvatureRotational 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 3d steady Gradient Ricci Solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2016)5Deng 2016 Level set flow in 3D steady gradient Ricci solitons (Chih-Wei Chen and Kuo-Wei Lee, 2017)8Chen 2017 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)1Lai 2022 3D flying wings for any asymptotic cones (Yi Lai, 2022)2Lai 2022 A family of Kähler flying wing steady Ricci solitons (Pak-Yeung Chan, Ronan J. Conlon and Yi Lai, 2024)3Chan 2024 Steady Gradient Ricci Solitons with O(p)× O(q) Symmetry (Lucas Lavoyer and Luke T. Peachey, 2025)7Lavoyer 2025 Higher-dimensional flying wing Steady Ricci Solitons (Pak-Yeung Chan, Yi Lai and Man-Chun Lee, 2025)4Chan 2025 Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture (Richard H. Bamler and Yi Lai, 2025)6Bamler 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. O(2)-symmetry of 3D steady gradient Ricci solitons

    Yi Lai

    Topics 3-manifolds, Ricci solitons · Same author Yi Lai · Shared terms “flying wing”, “steady gradient”, “positive curvature”

  2. 3D flying wings for any asymptotic cones

    Yi Lai

    Topics 3-manifolds, Ricci solitons · Same author Yi Lai · Shared terms “flying wing”, “steady gradient”, “gradient Ricci”

  3. A family of Kähler flying wing steady Ricci solitons

    Pak-Yeung Chan, Ronan J. Conlon and Yi Lai

    Topics Sphere theorems, Ricci solitons · Same author Yi Lai · Shared terms “flying wing”, “steady gradient”, “positive curvature”

  4. Higher-dimensional flying wing Steady Ricci Solitons

    Pak-Yeung Chan, Yi Lai and Man-Chun Lee

    Topics (Non)collapsing, Ricci solitons · Same author Yi Lai · Shared terms “flying wing”, “steady gradient”, “curvature operator”

  5. 3d steady Gradient Ricci Solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

    Topics Rotational symmetry & warped products, (Non)collapsing, 3-manifolds · Shared terms “steady gradient”, “dimensional steady”, “gradient Ricci”

  6. Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

    Richard H. Bamler and Yi LaiWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing · Same author Yi Lai · Shared terms “flying wing”, “collapsed”, “rotationally symmetric”

  7. Steady Gradient Ricci Solitons with Symmetry

    Lucas Lavoyer and Luke T. Peachey

    Topics Sphere theorems, Ricci solitons · Shared terms “steady gradient”, “positive curvature”, “gradient Ricci”

  8. Level set flow in 3D steady gradient Ricci solitons

    Chih-Wei Chen and Kuo-Wei Lee

    Topics 3-manifolds, Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “dimensional steady”, “gradient Ricci”

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.

Foundational paper in the Ricci solitons guide: Lai: flying-wing steady solitons, showing how rich the collapsed case is.

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) (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) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.