Generalized Lagrangian mean curvature flow in symplectic manifolds

Knut Smoczyk Leibniz University of Hanover Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10045

Asian Journal of Mathematics , 15, (1), 129-140, 2011
An almost K\"ahler 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 K\"ahler structure and we refer to (N,ω,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 $\hn$ 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 K\"ahler-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 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.
Lagrangian mean curvature flow, almost K¨ahler structure, symplectic manifold, cotangent bundle
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  title={Generalized Lagrangian mean curvature flow in symplectic manifolds},
  author={Knut Smoczyk, and Mu-Tao Wang},
  booktitle={Asian Journal of Mathematics },
Knut Smoczyk, and Mu-Tao Wang. Generalized Lagrangian mean curvature flow in symplectic manifolds. 2011. Vol. 15. In Asian Journal of Mathematics . pp.129-140.
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