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Let (M,J) be an almost complex manifold. We show that theinfinite-dimensional space Tau of totally real submanifolds in M carries anatural connection. This induces a canonical notion of geodesics in Tau and acorresponding definition of when a functional, defined on Tau, is convex.  Geodesics in Tau can be expressed in terms of families of J-holomorphiccurves in M; we prove a uniqueness result and study their existence. When M isK\"ahler we define a canonical functional on Tau; it is convex if M hasnon-positive Ricci curvature.  Our construction is formally analogous to the notion of geodesics and theMabuchi functional on the space of K\"ahler potentials, as studied byDonaldson, Fujiki and Semmes. Motivated by this analogy, we discuss possibleapplications of our theory to the study of minimal Lagrangians in negativeK\"ahler-Einstein manifolds.

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