Paper · Apr 9, 2024

On the shrinking solitons of generalized Ricci flow

Xilun Li, Yanan Ye

math.DGarXiv:2404.06141
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Abstract

We show that every gradient shrinking soliton of the generalized Ricci flow on compact manifold is a Ricci soliton. And we prove that the pluriclosed soliton is gradient Kahler-Ricci soliton under a broad cohomological condition. Moreover, we construct the first example of non-trivial shrinking generalized soliton, which can serve as a singularity model of the generalized Ricci flow.

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Submitted
Apr 9, 2024
Subjects
math.DG
Ricci flowGeneralized & G₂ flowsRicci solitonsKähler–Ricci solitonsSingularitiesKähler 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 2021, 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 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)8Apostolov 2021 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)6Bamler 2022 Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow (Mark Haskins, Ilyas Khan and Alec Payne, 2022)5Haskins 2022 Nash Entropy and Heat Kernel on Generalized Ricci Flow (Xilun Li, 2022)2Li 2022 Derivative estimates of pluriclosed flow (Yanan Ye, 2023)1Ye 2023 Dynamical stability of Pluriclosed and Generalized Ricci solitons (Kuan-Hui Lee, 2025)3Lee 2025 The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe (Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi, 2025)7Bunk 2025 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)4Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Derivative estimates of pluriclosed flow

    Yanan Ye

    Topics Generalized & G₂ flows, Kähler–Ricci solitons, Kähler geometry · Same author Yanan Ye · Shared terms “pluriclosed”

  2. Nash Entropy and Heat Kernel on Generalized Ricci Flow

    Xilun Li

    Topics Generalized & G₂ flows · Same author Xilun Li

  3. Dynamical stability of Pluriclosed and Generalized Ricci solitons

    Kuan-Hui Lee

    Topics Generalized & G₂ flows, Ricci solitons · Shared terms “pluriclosed”

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

    Ronan J. Conlon and Alix Deruelle

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

  5. Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow

    Mark Haskins, Ilyas Khan and Alec Payne

    Topics Generalized & G₂ flows, Ricci solitons · Shared terms “shrinking solitons”, “gradient shrinking”

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

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

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

  7. The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe

    Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi

    Topics Generalized & G₂ flows, Ricci solitons

  8. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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)

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  3. 6A.13
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