Paper · Nov 27, 2017

Coupled Kähler-Ricci solitons on toric Fano manifolds

Jakob Hultgren

math.DGarXiv:1711.09881v2
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Abstract

We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton type equations on toric Fano manifolds. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the Kähler-Ricci flow.

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Submitted
Nov 27, 2017
Last revised
Oct 5, 2018 (v2)
Subjects
math.DG, math.CV
Journal
Analysis & PDE 12 (2019) 2067-2094
DOI
10.2140/apde.2019.12.2067

Comments: 29 pages. Changed title. Added example of coupled Kähler-Einstein metric on manifold that don't admit Kähler-Einstein metrics

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

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 K-Stability of Fano spherical varieties (Thibaut Delcroix, 2016)4Delcroix 2016 Coupled Sasaki-Ricci solitons (Akito Futaki and Yingying Zhang, 2018)6Futaki 2018 Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds (Thibaut Delcroix and Jakob Hultgren, 2018)1Delcroix 2018 Examples of K-unstable Fano manifolds (Thibaut Delcroix, 2019)7Delcroix 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)3Guo 2020 Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold (Feng Wang and Xiaohua Zhu, 2020)5Wang 2020 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)8Nitta 2026 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)2Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds

    Thibaut Delcroix and Jakob Hultgren

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Jakob Hultgren · Shared terms “coupled Kähler”, “Fano”, “sufficient conditions”

  2. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “toric Fano”, “Fano manifolds”

  3. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Fano manifolds”

  4. K-Stability of Fano spherical varieties

    Thibaut Delcroix

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Fano manifolds”, “Donaldson conjecture”, “Tian Donaldson”

  5. Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Fano manifolds”, “Tian”

  6. Coupled Sasaki-Ricci solitons

    Akito Futaki and Yingying Zhang

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

  7. Examples of K-unstable Fano manifolds

    Thibaut Delcroix

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

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

    Yasufumi Nitta

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

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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

  3. 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.