2009/10/14 by Smoczyk, Knut, Wang, Mu-Tao
#53C44 #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.0910.2667
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(⋅, ⋅)=g(J(⋅), ⋅). 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 \citeb in Kähler manifolds that are almost Einstein.