Lagrangian submanifolds in strictly nearly K\"ahler 6-manifolds

Abstract

Lagrangian submanifolds in strict nearly K\"ahler 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 K\"ahler manifold $(M^{2n}, J, g)$ 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 K\"ahler manifold $(M^6, J, g)$. This formula implies, in particular, that any formal infinitesimal Lagrangian deformation of $L^3$ is a Jacobi field on $L^3$. We describe a finite dimensional local model of the moduli space of compact Lagrangian submanifolds in a strict nearly K\"ahler 6-manifold. We show that there is a real analytic atlas on $(M^6, J, g)$ in which the strict nearly K\"ahler structure $(J, g)$ is real analytic. Furthermore, w.r.t. an analytic strict nearly K\"ahler 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 K\"ahler structure described in \cite{Lotay2012}.