arXiv 0910.26672009-10-14math.DGmath.SG

辛流形中的广义Lagrange中曲率流

Generalized Lagrangian mean curvature flows in symplectic manifolds

Knut Smoczyk, Mu-Tao Wang

备注：: 15 pages

摘要：

辛流形$(N, ω)$上的几乎Kähler结构由一个黎曼度量$g$和一个几乎复结构$J$组成，使得辛形式$ω$满足$ω(\cdot, \cdot)=g(J(\cdot), \cdot)$。任何辛流形都有一个几乎凯勒结构，我们称$(N, ω, g, J)$为几乎凯勒流形。在这篇文章中，我们提出了一个自然的发展方程来研究几乎Kähler流形中的Lagrange子流形的变形。$TN$上的度量复联络$\hn$定义了沿$N$的任意拉格朗日子流形$M$的广义中曲率向量场。我们研究了$M$沿着这个矢量场的演化，只要连接$\hn$满足爱因斯坦条件，它就变成了一个拉格朗日变形。这可以看作是经典拉格朗日中曲率流在Kähler-Einstein流形中的推广，其中连接$\hn$是$g$的Levi-Civita连接。我们的结果适用于一个重要的情况下，拉格朗日子流形的余切丛配备了典型的几乎Kähler结构和其他推广的拉格朗日中曲率流，如流量考虑Behrndt \cite{b}在Kähler流形，几乎是爱因斯坦。

英文摘要：

An almost Kähler structure on a symplectic manifold $(N, ω)$ consists of a Riemannian metric $g$ and an almost complex structure $J$ such that the symplectic form $ω$ satisfies $ω(\cdot, \cdot)=g(J(\cdot), \cdot)$. Any symplectic manifold admits an almost Kähler structure and we refer to $(N, ω, g, J)$ as an almost Kähler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost Kähler manifolds. A metric and complex connection $\hn$ on $TN$ defines a generalized mean curvature vector field along any Lagrangian submanifold $M$ of $N$. We study the evolution of $M$ along this vector field, which turns out to be a Lagrangian deformation, as long as the connection $\hn$ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in Kähler-Einstein manifolds where the connection $\hn$ is the Levi-Civita connection of $g$. Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost Kähler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt \cite{b} in Kähler manifolds that are almost Einstein.

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