Paper · Jan 30, 2017

Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones

Brett Kotschwar

math.DGarXiv:1701.08486v2
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As on arXiv

Abstract

We prove that a shrinking gradient Ricci soliton which is asymptotic to a Kähler cone along some end is itself Kähler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally Kähler.

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Submitted
Jan 30, 2017
Last revised
Dec 7, 2017 (v2)
Subjects
math.DG
DOI
10.1007/s12220-017-9922-0

Comments: v2: 13 pages, minor changes. To appear in J. Geom. Anal

Ricci solitonsKähler geometry

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 A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons (Brett Kotschwar and Lu Wang, 2017)3Kotschwar 2017 Isometries of asymptotically conical shrinking Ricci solitons (Brett Kotschwar and Lu Wang, 2018)2Kotschwar 2018 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)8Bamler 2022 Ricci flows which terminate in cones (Brett Kotschwar, 2023)4Kotschwar 2023 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)5Bamler 2026 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)1Conlon 2026 Kählerity of complete almost-Kähler gradient shrinking Ricci solitons (Junming Xie, 2026)6Xie 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)7Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler geometry, Ricci solitons · Shared terms “Kähler cone”, “shrinking gradient”

  2. Isometries of asymptotically conical shrinking Ricci solitons

    Brett Kotschwar and Lu Wang

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

  3. A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

    Brett Kotschwar and Lu Wang

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

  4. Ricci flows which terminate in cones

    Brett Kotschwar

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

  5. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon

    Topics Kähler geometry, Ricci solitons · Shared terms “Kähler cone”, “gradient Ricci”

  6. Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

    Junming Xie

    Topics Kähler geometry, Ricci solitons · Shared terms “kahlerity”

  7. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler geometry, Ricci solitons · Shared terms “shrinking gradient”

  8. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al.

    Topics Kähler geometry, Ricci solitons · Shared terms “shrinking gradient”

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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.