Paper · Dec 8, 2017

A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

Brett Kotschwar, Lu Wang

math.DGarXiv:1712.03185v2
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Abstract

We prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with one of the standard cylindrical metrics on for along some end must be isometric to the cylinder on that end. When the underlying manifold is complete, it must be globally isometric either to the cylinder or (when ) to its -quotient.

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Submitted
Dec 8, 2017
Last revised
Sep 15, 2020 (v2)
Subjects
math.DG

Comments: v2: Minor corrections; 66 pages, no figures

Ricci solitonsStability & uniqueness

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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 Structure at infinity for shrinking Ricci solitons (Ovidiu Munteanu and Jiaping Wang, 2016)6Munteanu 2016 Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones (Brett Kotschwar, 2017)2Kotschwar 2017 Ricci flow from some spaces with asymptotically cylindrical singularities (Timothy Carson, 2018)7Carson 2018 Isometries of asymptotically conical shrinking Ricci solitons (Brett Kotschwar and Lu Wang, 2018)1Kotschwar 2018 Uncertainty Principle and its rigidity on complete gradient shrinking Ricci solitons (Weixiong Mai and Jianyu Ou, 2019)4Mai 2019 Ricci flows which terminate in cones (Brett Kotschwar, 2023)3Kotschwar 2023 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)8Kotschwar 2023 On steady and expanding Ricci solitons with asymptotic symmetries (Michael B. Law, 2025)5Law 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Isometries of asymptotically conical shrinking Ricci solitons

    Brett Kotschwar and Lu Wang

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

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

    Brett Kotschwar

    Topics Ricci solitons · Same author Brett Kotschwar · Shared terms “shrinking gradient”, “gradient Ricci”, “globally”

  3. Ricci flows which terminate in cones

    Brett Kotschwar

    Topics Ricci solitons · Same author Brett Kotschwar

  4. Uncertainty Principle and its rigidity on complete gradient shrinking Ricci solitons

    Weixiong Mai and Jianyu Ou

    Topics Stability & uniqueness, Ricci solitons · Shared terms “shrinking Ricci”, “shrinking gradient”, “gradient Ricci”

  5. On steady and expanding Ricci solitons with asymptotic symmetries

    Michael B. Law

    Topics Stability & uniqueness, Ricci solitons · Shared terms “asymptotically cylindrical”, “quotient”, “gradient Ricci”

  6. Structure at infinity for shrinking Ricci solitons

    Ovidiu Munteanu and Jiaping Wang

    Topics Ricci solitons · Shared terms “shrinking Ricci”, “quotient”

  7. Ricci flow from some spaces with asymptotically cylindrical singularities

    Timothy Carson

    Topics Stability & uniqueness · Shared terms “asymptotically cylindrical”, “globally”

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

    Brett Kotschwar

    Topics Ricci solitons · Same author Brett Kotschwar

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.