An \(\varepsilon \)-regularity theorem for line bundle mean curvature flow (Q6160103)

The line bundle mean curvature flow was introduced by \textit{A. Jacob} and \textit{S.-T. Yau} [Math. Ann. 369, No. 1--2, 869--898 (2017; Zbl 1375.32045)] as the negative gradient flow of a volume functional on the space of Hermitian metrics on a holomorphic line bundle \(L\to X\) over a Kähler manifold \((X,g)\). As the minimizers of the volume functional are deformed Hermitian Yang-Mills (dHYM) metrics, the line bundle mean curvature flow can be used to find the dHYM metrics. Some sufficient conditions were provided in [loc. cit.] to ensure that a line bundle mean curvature flow \(h_t\) defined for \(t\in [0,T)\) can be extended beyond the time \(T\). In this paper, the authors prove an \(\varepsilon\)-regularity theorem for the line bundle mean curvature flow. This is motivated by the work of \textit{B. White} [Ann. Math. (2) 161, No. 3, 1487--1519 (2005; Zbl 1091.53045)] on the \(\varepsilon\)-regularity of mean curvature flows. For the proof, the authors discover a monotonicity formula and introduce a Gaussian density for line bundle mean curvature flows. This is in analogy with a result in [\textit{G. Huisken}, J. Differ. Geom. 31, No. 1, 285--299 (1990; Zbl 0694.53005)] for mean curvature flows. The authors also introduce the definition of self-shrinkers for line bundle mean curvature flows and prove a Liouville-type theorem, which is used crucially in the proof of the \(\varepsilon\)-regularity theorem.