Minimal Lagrangian submanifolds in indefinite complex space

TL;DR

This paper studies minimal Lagrangian submanifolds in indefinite complex space, showing they minimize volume in their homology class and describing various families, including surfaces in C^2 and equivariant submanifolds.

Contribution

It characterizes minimal Lagrangian submanifolds in indefinite complex space and provides explicit descriptions of several families, advancing understanding of their geometric properties.

Findings

Minimal Lagrangian submanifolds minimize volume in their homology class.

Characterization of minimal Lagrangian surfaces in C^2 with neutral metric.

Description of equivariant minimal Lagrangian submanifolds in C^n with arbitrary signature.

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.