Inserted: 16 apr 2014
We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by the ambient Ricci curvature or by analogues in non-Kaehler manifolds. The Streets--Tian symplectic curvature flow is shown to play an important role in this context, thus emphasizing the possible interest in almost Kaehler manifolds for geometric analysis. We develop new geometric quantities for totally real submanifolds using the ambient canonical bundle KM. In particular we develop a notion of geodesics, related to pseudo-holomorphic curves, and prove that our main functional is convex with respect to this notion. This may have applications to the study of Lagrangian submanifolds and calibrated geometry.