Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow
Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow
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DOI：
10.1016/j.geomphys.2018.02.011
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发表时间：
2017-10
影响因子：
1.5
通讯作者：
T. Kajigaya;Keita Kunikawa
中科院分区：
文献类型：
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作者：
T. Kajigaya;Keita Kunikawa
利用Behrndt（2011）提出的方法，将Kähler-Einstein流形中Lagrange子流形的Hamilton稳定性和平均曲率流的几个结果推广到更一般的Kähler流形，包括具有Kähler形式ω∈ 2 π c1（M）的Fano流形.也就是说，我们首先考虑Kähler流形M中的Lagrange子流形L上的加权测度，并研究L对于加权体积泛函的变分问题。我们称加权体积泛函的一个不动点为f-极小，并将Hamilton f-稳定性定义为Hamilton变形下的局部极小。我们显示这样的例子自然出现在复曲面Fano流形。此外，我们考虑了Behrndt和Smoczyk-Wang提出的Fano流形中的广义Lagrange平均曲率流。推广了H. Li，证明了如果初始Lagrange子流形是f-极小且Hamilton f-稳定的Lagrange子流形的小Hamilton变形，则广义MCF指数快速收敛到f-极小Lagrange子流形.
In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler–Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form ω∈ 2 π c 1 (M) by using the method proposed by Behrndt (2011). Namely, we first consider a weighted measure on a Lagrangian submanifold L in a Kähler manifold M and investigate the variational problem of L for the weighted volume functional. We call a stationary point of the weighted volume functional f-minimal, and define the notion of Hamiltonian f-stability as a local minimizer under Hamiltonian deformations. We show such examples naturally appear in a toric Fano manifold. Moreover, we consider the generalized Lagrangian mean curvature flow in a Fano manifold which is introduced by Behrndt and Smoczyk–Wang. We generalize the result of H. Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian deformation of an f-minimal and Hamiltonian f-stable Lagrangian submanifold, then the generalized MCF converges exponentially fast to an f-minimal Lagrangian submanifold.