Paper · Jan 21, 2020

Stability of line bundle mean curvature flow

Xiaoli Han, Xishen Jin

math.DGWider flowsarXiv:2001.07406v2
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Abstract

Let be a compact Kähler manifold of complex dimension and be a holomorphic line bundle over . The line bundle mean curvature flow was introduced in in order to find deformed Hermitian-Yang-Mills metrics on . In this paper, we consider the stability of the line bundle mean curvature flow. Suppose there exists a deformed Hermitian Yang-Mills metric on . We prove that the line bundle mean curvature flow converges to exponentially in sense as long as the initial metric is close to in -norm.

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Submitted
Jan 21, 2020
Last revised
Feb 3, 2020 (v2)
Subjects
math.DG

Comments: Minor corrections in the proof of Theorem 1.5 on pp 22

Mean curvature flowKähler geometryStability & uniqueness

Its neighbourhood

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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 Positivity preserving along a flow over projective bundle (Xueyuan Wan, 2018)6Wan 2018 An ε-regularity theorem for line bundle mean curvature flow (Xiaoli Han and Hikaru Yamamoto, 2019)1Han 2019 Collapsing of the line bundle mean curvature flow on Kähler surfaces (Ryosuke Takahashi, 2019)2Takahashi 2019 Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow (Ryosuke Takahashi, 2020)7Takahashi 2020 Mirror of volume functionals on manifolds with special holonomy (Kotaro Kawai and Hikaru Yamamoto, 2021)4Kawai 2021 Singularity formation along the line bundle mean curvature flow (Yu Hin Chan and Adam Jacob, 2023)3Chan 2023 Stability of the generalized Lagrangian mean curvature flow in cotangent bundle (Xishen Jin and Jiawei Liu, 2024)5Jin 2024 Special Lagrangian smoothings, Calabi ansatz and stability conditions (Jacopo Stoppa, 2026)8Stoppa 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An -regularity theorem for line bundle mean curvature flow

    Xiaoli Han and Hikaru YamamotoWider flows

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

  2. 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”

  3. 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”

  4. Mirror of volume functionals on manifolds with special holonomy

    Kotaro Kawai and Hikaru YamamotoWider flows

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

  5. Stability of the generalized Lagrangian mean curvature flow in cotangent bundle

    Xishen Jin and Jiawei LiuWider flows

    Topics Stability & uniqueness · Same author Xishen Jin

  6. Positivity preserving along a flow over projective bundle

    Xueyuan Wan

    Topics Kähler geometry · Shared terms “Hermitian Yang”, “Yang Mills”, “line bundle”

  7. Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow

    Ryosuke TakahashiWider flows

    Topics Mean curvature flow · Shared terms “bundle mean”, “deformed Hermitian”, “Hermitian Yang”

  8. Special Lagrangian smoothings, Calabi ansatz and stability conditions

    Jacopo StoppaWider flows

    Topics Kähler geometry, Stability & uniqueness · Shared terms “deformed Hermitian”, “Hermitian Yang”, “Yang Mills”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.