Complexified diffeomorphism groups, totally real submanifolds and K\"ahler-Einstein geometry

Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space Tau of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in Tau and a corresponding definition of when a functional, defined on Tau, is convex. Geodesics in Tau can be expressed in terms of families of J-holomorphic curves in M; we prove a uniqueness result and study their existence. When M is K\"ahler we define a canonical functional on Tau; it is convex if M has non-positive Ricci curvature. Our construction is formally analogous to the notion of geodesics and the Mabuchi functional on the space of K\"ahler potentials, as studied by Donaldson, Fujiki and Semmes. Motivated by this analogy, we discuss possible applications of our theory to the study of minimal Lagrangians in negative K\"ahler-Einstein manifolds.