Minimal lagrangian submanifolds of Kähler-einstein manifolds

Abstract

An interesting class of submanifolds of a Kähler manifold M²ⁿ is the class of submanifolds Nⁿ ⊑ M²ⁿ which are minimal with respect to the metric on M²ⁿ and are Lagrangian with respect to the symplectic form on M²ⁿ. A general Kähler manifold will not have any of these submanifolds. However, in this paper, we show that if the metric on M²ⁿ is also Einstein, then these minimal Lagrangian submanifolds exist in abundance, at least locally. We give a precise description of this "generality" in terms of Cartan-Kähler theory and relate these submanifolds to the calibrated geometries of Harvey and Lawson and to maximal real structures on algebraic varieties.