From Lagrangian to totally real geometry: coupled flows and calibrations

Abstract

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 ambient Ricci curvature or, in the non-K\"ahler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss short-time existence for our flow and show it couples well with the Streets-Tian symplectic curvature flow for almost K\"ahler manifolds. We also discuss possible applications to Lagrangian submanifolds and calibrated geometry.