Paper · Apr 4, 2019

An -regularity theorem for line bundle mean curvature flow

Xiaoli Han, Hikaru Yamamoto

math.DGWider flowsarXiv:1904.02391v3
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Abstract

In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kähler manifold. The goal of this paper is to give an -regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone quantity. As a critical point of this quantity, we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers is also given. It plays an important role in the proof of the -regularity theorem.

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Submitted
Apr 4, 2019
Last revised
Jul 24, 2019 (v3)
Subjects
math.DG

Comments: 41 pages. v2: some comments were added in the introduction. v3: some comments were added in the information of funds

Mean curvature flowSelf-shrinkers & translatorsKähler 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 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 Examples of Ricci-mean curvature flows (Hikaru Yamamoto, 2016)5Yamamoto 2016 Collapsing of the line bundle mean curvature flow on Kähler surfaces (Ryosuke Takahashi, 2019)3Takahashi 2019 Stability of line bundle mean curvature flow (Xiaoli Han and Xishen Jin, 2020)1Han 2020 Mirror of volume functionals on manifolds with special holonomy (Kotaro Kawai and Hikaru Yamamoto, 2021)2Kawai 2021 Singularity formation along the line bundle mean curvature flow (Yu Hin Chan and Adam Jacob, 2023)4Chan 2023 Gap theorems for complete self-shrinkers of r-mean curvature flows (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2024)7Alencar 2024 Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class (Xiaoli Han, Jiayu Li and Jun Sun, 2024)6Han 2024 A gap Theorem on closed self-shrinkers of mean curvature flow (Yuhang Zhao, 2025)8Zhao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of line bundle mean curvature flow

    Xiaoli Han and Xishen JinWider flows

    Topics Kähler geometry · Same author Xiaoli Han · Shared terms “bundle mean”, “line bundle”, “deformed Hermitian”

  2. Mirror of volume functionals on manifolds with special holonomy

    Kotaro Kawai and Hikaru YamamotoWider flows

    Topics Kähler geometry · Same author Hikaru Yamamoto · Shared terms “line bundle”, “bundle mean”, “deformed Hermitian”

  3. Collapsing of the line bundle mean curvature flow on Kähler surfaces

    Ryosuke TakahashiWider flows

    Topics Kähler geometry · Shared terms “bundle mean”, “line bundle”, “deformed Hermitian”

  4. Singularity formation along the line bundle mean curvature flow

    Yu Hin Chan and Adam JacobWider flows

    Topics Kähler geometry · Shared terms “bundle mean”, “line bundle”, “deformed Hermitian”

  5. Examples of Ricci-mean curvature flows

    Hikaru Yamamoto

    Topics Self-shrinkers & translators · Same author Hikaru Yamamoto

  6. Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class

    Xiaoli Han, Jiayu Li and Jun SunWider flows

    Topics Self-shrinkers & translators · Same author Xiaoli Han

  7. Gap theorems for complete self-shrinkers of -mean curvature flows

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows

    Topics Self-shrinkers & translators · Shared terms “Yau”

  8. A gap Theorem on closed self-shrinkers of mean curvature flow

    Yuhang ZhaoWider flows

    Topics Self-shrinkers & translators

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.