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We introduce the notion of K\"ahler manifolds that are almost Einstein and wedefine a generalized mean curvature vector field along submanifolds in them. Weprove that Lagrangian submanifolds remain Lagrangian, when deformed indirection of the generalized mean curvature vector field. For a K\"ahlermanifold that is almost Einstein, and which in addition has a trivial canonicalbundle, we show that the generalized mean curvature vector field of aLagrangian submanifold is the dual vector field associated to the Lagrangianangle.

Generalized Lagrangian mean curvature flow in K\"ahler manifolds that are almost Einstein