Generalized Lagrangian mean curvature flows in symplectic manifolds

Abstract

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.