Paper · Aug 21, 2017

Toric extremal Kaehler-Ricci solitons are Kaehler-Einstein

Simone Calamai, David Petrecca

math.DGarXiv:1708.06256v2
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Abstract

In this short note, we prove that a Calabi extremal Kaehler-Ricci soliton on a compact toric Kaehler manifold is Einstein. This solves for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.

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Submitted
Aug 21, 2017
Last revised
Sep 5, 2017 (v2)
Subjects
math.DG

Comments: 4 pages. Some changes in the references, modified introduction and misprints corrected. Comments are welcome

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stability

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On canonical radial Kaehler metrics

    Andrea Loi, Filippo Salis and Fabio Zuddas

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Kaehler”, “extremal”

  2. Kaehler Ricci solitons induced by infinite dimensional complex space forms

    Andrea Loi, Fabio Zuddas and Filippo Salis

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Kaehler Ricci”, “Kaehler Einstein”

  3. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “extremal”

  4. Irregular Eguchi-Hanson type metrics and their soliton analogues

    Akito Futaki

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Kaehler Einstein”, “Kaehler Ricci”

  5. A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons

    Huai-Dong Cao, Xiaofeng Sun and Yingying Zhang

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Kaehler Ricci”

  6. Deformations of Fano manifolds with weighted solitons

    Akito Futaki

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Kaehler”, “Calabi”

  7. On weighted extremal Kähler metrics

    Akito Futaki

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “extremal”, “Calabi”

  8. An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics

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–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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.