Paper · Jul 15, 2019

Kähler-Ricci flow on horospherical manifold

François Delgove

math.DGarXiv:1907.07112
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Abstract

In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.

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Submitted
Jul 15, 2019
Subjects
math.DG

Comments: arXiv admin note: text overlap with arXiv:1710.05623

Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsKähler geometry

Its neighbourhood

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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 Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons (James Isenberg, Dan Knopf and Natasa Sesum, 2017)6Isenberg 2017 Kähler-Ricci solitons on some wonderful group compactifications (François Delgove, 2017)2Delgove 2017 Kähler-Ricci solitons on horospherical variety (François Delgove, 2017)1Delgove 2017 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)3Guo 2020 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)4Tian 2022 A Note On Kähler-Ricci Flow on Fano Threefolds (Minghao Miao and Gang Tian, 2022)8Miao 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)5Wang 2023 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)7Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci solitons on horospherical variety

    François Delgove

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author François Delgove · Shared terms “horospherical”

  2. Kähler-Ricci solitons on some wonderful group compactifications

    François Delgove

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author François Delgove

  3. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

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

  4. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Shared terms “Fano”, “Gromov”

  5. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Shared terms “Fano”, “Gromov”

  6. Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons

    James Isenberg, Dan Knopf and Natasa Sesum

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry

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

    Liding Huang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Shared terms “Fano”, “flow converges”

  8. A Note On Kähler-Ricci Flow on Fano Threefolds

    Minghao Miao and Gang Tian

    Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Shared terms “Fano”, “Gromov”

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.