Asian Journal of Mathematics

Volume 26 (2022)

Number 6

An $\varepsilon$-regularity theorem for line bundle mean curvature flow

Pages: 737 – 776



Xiaoli Han (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Hikaru Yamamoto (Department of Mathematics, Faculty of Pure and Applied Science, University of Tsukuba, Ibaraki, Japan)


In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau [7]. 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 $\varepsilon$-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 $\varepsilon$-regularity theorem.

2010 Mathematics Subject Classification

53C38, 53D37

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The first author was supported by NSFC, No. 12031017.

The second author was supported by JSPS KAKENHI Grant Number 18K13415, and by and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).

Received 15 January 2021

Accepted 17 August 2022

Published 27 April 2023