Paper · Dec 31, 2018

Isometries of asymptotically conical shrinking Ricci solitons

Brett Kotschwar, Lu Wang

math.DGarXiv:1901.00044
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Abstract

We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.

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Submitted
Dec 31, 2018
Subjects
math.DG

Comments: 12 pages

Ricci solitonsBoundaries & 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 Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones (Brett Kotschwar, 2017)4Kotschwar 2017 A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons (Brett Kotschwar and Lu Wang, 2017)1Kotschwar 2017 The space of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2017)5Bernstein 2017 Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2018)8Bernstein 2018 Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow (Mark Haskins, Ilyas Khan and Alec Payne, 2022)3Haskins 2022 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)2Kotschwar 2023 Ricci flows which terminate in cones (Brett Kotschwar, 2023)6Kotschwar 2023 Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons (Carlos Esparza, 2025)7Esparza 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

    Brett Kotschwar and Lu Wang

    Topics Ricci solitons · Same authors Brett Kotschwar, Lu Wang · Shared terms “shrinking Ricci”

  2. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Boundaries & non-compact, Ricci solitons · Same author Brett Kotschwar · Shared terms “asymptotically conical”, “cross section”, “shrinking soliton”

  3. Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow

    Mark Haskins, Ilyas Khan and Alec Payne

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “asymptotically conical”, “shrinking soliton”, “shrinking Ricci”

  4. Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones

    Brett Kotschwar

    Topics Ricci solitons · Same author Brett Kotschwar · Shared terms “cone along”

  5. The space of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

    Topics Boundaries & non-compact · Same author Lu Wang · Shared terms “asymptotically conical”

  6. Ricci flows which terminate in cones

    Brett Kotschwar

    Topics Ricci solitons · Same author Brett Kotschwar · Shared terms “shrinking soliton”

  7. Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons

    Carlos Esparza

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “conical shrinking”, “asymptotically conical”

  8. Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

    Topics Boundaries & non-compact · Same author Lu Wang · Shared terms “asymptotically conical”

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 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 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.