Paper · Sep 1, 2021

Convergence of Hermitian manifolds and the Type IIB flow

Nikita Klemyatin

math.DGarXiv:2109.00312
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Abstract

The Type IIB flow is a flow of conformally balanced complex manifolds introduced by Phong, Picard, and Zhang, about whose singularities little is as yet known. We formulate convergence criteria for the Gromov-Cheeger-Hamilton convergence of sequences of Hermitian manifolds, and apply them to precompactness theorems and the existence of singularity models for the Type IIB flow, in analogy with Hamilton's classic compactness theorems and classification of singularities for the Ricci flow.

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Submitted
Sep 1, 2021
Subjects
math.DG, math.AP, math.CV

Comments: 22 pages

Ricci flowSingularitiesLimit spaces & RCD

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 Lower bounds for the scalar curvatures of Ricci flow singularity models (Pak-Yeung Chan, Bennett Chow, Zilu Ma et al., 2022)1Chan 2022 Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2022)6Bourni 2022 Ricci flow from spaces with edge type conical singularities (Lucas Lavoyer, 2023)3Lavoyer 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)4Hallgren 2023 Splitting maps in Type I Ricci flows (Panagiotis Gianniotis, 2024)5Gianniotis 2024 On the structure of noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)7Fang 2025 Diameter bounds in 3d Type I Ricci flows (Panagiotis Gianniotis, 2025)2Gianniotis 2025 Long-Time Behavior of the α-Ricci Flow on the 2-Torus (Yifan Guo and Xiaokang Wang, 2026)8Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Lower bounds for the scalar curvatures of Ricci flow singularity models

    Pak-Yeung Chan, Bennett Chow, Zilu Ma et al.

    Topics Limit spaces & RCD, Singularities · Shared terms “singularity models”, “compactness theorem”, “Cheeger”

  2. Diameter bounds in 3d Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, Singularities · Shared terms “Cheeger”

  3. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer

    Topics Limit spaces & RCD, Singularities · Shared terms “Gromov”

  4. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Limit spaces & RCD, Singularities · Shared terms “Gromov”, “sequence”

  5. Splitting maps in Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, Singularities · Shared terms “Cheeger”

  6. Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics Limit spaces & RCD · Shared terms “Hamilton”, “compactness”

  7. On the structure of noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li

    Topics Limit spaces & RCD · Shared terms “compactness theorem”, “Gromov”

  8. Long-Time Behavior of the -Ricci Flow on the 2-Torus

    Yifan Guo and Xiaokang Wang

    Topics Limit spaces & RCD · Shared terms “Gromov”, “sequence”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.