Lagrangian submanifolds in strict nearly Kähler 6-manifolds

Abstract

Lagrangian submanifolds in strict nearly Kahler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold $L$ in a nearly Kahler manifold $(M, J, g)$ is symplectically dual to the Maslov 1-form on $L$. Using relative calibrations, we derive a formula for the second variation of the volume of a Lagrangian submanifold $L^3$ in a strict nearly Kahler manifold $(M^6, J, g)$ and compare it with McLean's formula for special Lagrangian submanifolds. We describe a finite dimensional local model of the moduli space of compact Lagrangian submanifolds in a strict nearly Kahler 6-manifold. We show that there is a real analytic atlas on $(M^6, J, g)$ in which the strict nearly Kahler structure $(J, g)$ is real analytic. Furthermore, w.r.t. an analytic strict nearly Kahler structure the moduli space of Lagrangian submanifolds of $M^6$ is a real analytic variety, whence infinitesimal Lagrangian deformations are smoothly obstructed if and only if they are formally obstructed. As an application, we relate our results to the description of Lagrangian submanifolds in the sphere $S^6$ with the standard nearly Kahler structure described in [34].