Complexified diffeomorphism groups, totally real submanifolds and Kähler–Einstein geometry

Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space T of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in T and a corresponding definition of when a functional on T is convex. Geodesics in T 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 Kähler we define a canonical functional on T; 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 Kähler 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 Kähler-Einstein manifolds.