Lotay, JD; Pacini, T; (2019) Complexified diffeomorphism groups, totally real submanifolds and KÄhler–Einstein geometry. Transactions of the American Mathematical Society , 371 (4) pp. 2665-2701. 10.1090/tran/7421.

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Abstract

Let (M, J) be an almost complex manifold. We show that the infinitedimensional 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 f : T → R 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 K¨ahler 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 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.

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