Paper · Dec 31, 2019

Collapsing of the line bundle mean curvature flow on Kähler surfaces

Ryosuke Takahashi

math.DGWider flowsarXiv:1912.13145v4
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Abstract

We study the line bundle mean curvature flow on Kähler surfaces under the hypercritical phase and a certain semipositivity condition. We naturally encounter such a condition when considering the blowup of Kähler surfaces. We show that the flow converges smoothly to a singular solution to the deformed Hermitian-Yang-Mills equation away from a finite number of curves of negative self-intersection on the surface. As an application, we obtain a lower bound of a Kempf-Ness type functional on the space of potential functions satisfying the hypercritical phase condition.

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Submitted
Dec 31, 2019
Last revised
Jan 7, 2021 (v4)
Subjects
math.DG

Comments: 19 pages, final version, to appear in Calc. Var. Partial Differential Equations

Mean curvature flowSingularities(Non)collapsingKähler geometryLong-time existence & convergence

Its neighbourhood

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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 · Shared terms “bundle mean”, “line bundle”, “deformed Hermitian”

  2. Singularity formation along the line bundle mean curvature flow

    Yu Hin Chan and Adam JacobWider flows

    Topics Long-time existence & convergence, Kähler geometry, Singularities · Shared terms “bundle mean”, “line bundle”, “Mills equation”

  3. An -regularity theorem for line bundle mean curvature flow

    Xiaoli Han and Hikaru YamamotoWider flows

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

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

    Ryosuke TakahashiWider flows

    Topics Long-time existence & convergence · Same author Ryosuke Takahashi · Shared terms “bundle mean”, “line bundle”, “deformed Hermitian”

  5. Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Ronan J. Conlon, Max Hallgren and Zilu Ma

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “Kähler surfaces”, “line bundle”

  6. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al.

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “Kähler surfaces”

  7. Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

    Farnaz Ghanbari and SamreenaWider flows

    Topics (Non)collapsing, Kähler geometry, Singularities

  8. Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

    Tongxin Xu and Zhenlei Zhang

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “Kähler surfaces”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.