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MiS Preprint
57/2015

Lagrangian submanifolds in strict nearly Kähler 6-manifolds

Hông Vân Lê and Lorenz J. Schwachhöfer

Abstract

Lagrangian submanifolds in strict nearly Kähler 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 Kähler 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 Kähler 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 Kähler 6-manifold. We show that there is a real analytic atlas on $(M^6, J, g)$ in which the strict nearly Kähler structure $(J, g)$ is real analytic. Furthermore, w.r.t. an analytic strict nearly Kähler 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 Kähler structure described in: J. Lotay, Stability of coassociative conical singularities, Comm. Anal. Geom. 20 (2012), no. 4, 803867.

Received:
Sep 14, 2015
Published:
Sep 21, 2015
MSC Codes:
53C40, 53C38, 53D12, 58D99
Keywords:
nearly Kähler 6-manifold, Lagrangian submanifold, calibration, Jacobi field, moduli space

Related publications

inJournal
2019 Journal Open Access
Hông Vân Lê and Lorenz J. Schwachhöfer

Lagrangian submanifolds in strict nearly Kähler 6-manifolds

In: Osaka journal of mathematics, 56 (2019) 3, pp. 601-629