Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds

摘要

Given an almost para-Kähler manifold equipped with a metric and para-complex connection, we define a generalized second fundamental form and generalized mean curvature vector of space-like Lagrangian submanifolds. We then show that the deformation induced by this variant of the mean curvature vector field preserves the Lagrangian condition, if the connection satisfies also some Einstein condition. In case the almost para-Kähler structure is integrable, the flow coincides with the classical mean curvature flow in the pseudo-Riemannian context.