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Singularities of Lagrangian mean curvature flow: zero-Maslov class case. (English) Zbl 1119.53052

The paper is concerned with singularities of Lagrangian mean curvature flow in \(\mathbb C^n\) such that the initial condition is a zero-Maslov class Lagrangian. It contains two main results: Theorem A states that the tangent flow at a singularity can be decomposed into a finite union of area-minimizing Lagrangian cones. This is an extension of previous results by M.-T. Wang and J. Chen and J. Li. Theorem B states that when the initial condition is an almost-calibrated and rational Lagrangian, the Lagrangian angle converges to a single constant on each connected component of the rescaled flow. Hence connected components of the rescaled flow converge weakly to a single area-minimizing Lagrangian cone. Some examples of zero-Maslov class exact Lagrangians which develop finite-time singularities under Lagrangian mean curvature flow are presented. These examples include Lagrangians with arbitrarily small oscillation of the Lagrangian angle and Lagrangians which are Hamiltonian isotopic to a plane.

MSC:

53D12 Lagrangian submanifolds; Maslov index
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