Minimal Lagrangian submanifolds in indefinite complex space

Abstract

Consider the complex linear space C^n endowed with the canonical pseudo-Hermitian form of signature (2p,2(n-p)). This yields both a pseudo-Riemannian and a symplectic structure on C^n. We prove that those submanifolds which are both Lagrangian and minimal with respect to these structures minimize the volume in their Lagrangian homology class. We also describe several families of minimal Lagrangian submanifolds. In particular, we characterize the minimal Lagrangian surfaces in C^2 endowed with its natural neutral metric and the equivariant minimal Lagrangian submanifolds of C^n with arbitrary signature.