Generalized Lagrangian mean curvature flows in symplectic manifolds

An almost K"ahler structure on a symplectic manifold $(N,\omega )$ consists of a Riemannian metric $g$ and an almost complex structure $J$ such that the symplectic form $\omega$ satisfies $\omega (\cdot ,\cdot )=g(J(\cdot ),\cdot )$. Any symplectic manifold admits an almost K"ahler structure and we refer to $(N,\omega ,g,J)$ as an almost K"ahler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost K"ahler manifolds. A metric and complex connection $\text{hn}$ 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 $\text{hn}$ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in K"ahler-Einstein manifolds where the connection $\text{hn}$ 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 K"ahler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt \cite{b} in K"ahler manifolds that are almost Einstein.