Collapsing of the line bundle mean curvature flow on Kähler surfaces

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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Notes

  1. 1.

    In [14], they made an assumption that the form F arises as curvature of a fiber metric on a holomorphic line bundle. But this assumption is only for aesthetic purposes and not essentially used in their paper.

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Correspondence to Ryosuke Takahashi.

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Takahashi, R. Collapsing of the line bundle mean curvature flow on Kähler surfaces. Calc. Var. 60, 27 (2021). https://doi.org/10.1007/s00526-020-01908-0

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Mathematics Subject Classification

  • 53C55
  • 53C44