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Generalized Lagrangian mean curvature flows in almost Calabi-Yau manifolds. (English) Zbl 1372.53068

In this paper, the authors study the generalized Lagrangian mean curvature flow (MCF) in almost Einstein manifolds which was proposed by T. Behrndt [Springer Proc. Math. 8, 65–79 (2011; Zbl 1243.53106)]. There are two results in this paper.
The first one is a long-time existence theorem, which states that if the second fundamental form of the flowing Lagrangian submanifold is uniformly bounded along the generalized Lagrangian MCF in the time interval \([0,T)\), then the flow can be extended smoothly past time \(T\). In other words, the singularities of the generalized Lagrangian MCF are characterized by the second fundamental form. The proof is done by applying the maximum principle to the evolution equations of the second fundamental form and its higher-order derivatives to deduce that the derivatives of the velocity of the flow is bounded once the second fundamental form is bounded, then the flow can be extended smoothly by standard argument. By this characterization, the singularities of the flow can be divided into two types in terms of the blow up rate, similar as in Huisken’s work on MCF in Euclidean space.
The second purpose of this paper is to generalize A. Neves’ [Invent. Math. 168, No. 3, 449–484 (2007; Zbl 1119.53052)] result on Lagrangian MCF in Calabi-Yau manifolds with zero Maslov class to the generalized Lagrangian MCF in almost Calabi-Yau manifolds. By following a similar argument as in Neves’ paper, the authors show that any rescaled flow at a singularity converges to a finite union of special Lagrangian cones for generalized Lagrangian MCF with zero Maslov class in almost Calabi-Yau manifolds. In particular, the corollary of this result is that there is no finite type-I singularity for such flow.
Reviewer: Yong Wei (Act)

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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References:

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