Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein

Abstract

We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.

13 pages. Former title: 'Lagrangian mean curvature flow in almost Kähler-Einstein manifolds'