Lagrangian submanifolds in strict nearly Kähler 6-manifolds

Hông Vân Lê and Lorenz J. Schwachhöfer

Contact the author: Please use for correspondence this email.
Submission date: 14. Sep. 2015
Pages: 34
published in: Osaka journal of mathematics, 56 (2019) 3, p. 601-629 
Bibtex
MSC-Numbers: 53C40, 53C38, 53D12, 58D99
Keywords and phrases: nearly Kähler 6-manifold, Lagrangian submanifold, calibration, Jacobi field, moduli space
Download full preprint: PDF (359 kB)

Abstract:
Lagrangian submanifolds in strict nearly Kähler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in G₂-manifolds. We prove that the mean curvature of a Lagrangian submanifold L in a nearly Kähler manifold (M²ⁿ,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 L³ in a strict nearly Kähler manifold (M⁶,J,g). This formula implies, in particular, that any formal infinitesimal Lagrangian deformation of L³ is a Jacobi field on L³. 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 (M⁶,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 M⁶ 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 S⁶ with the standard nearly Kähler structure described in: J. Lotay, Stability of coassociative conical singularities, Comm. Anal. Geom. 20 (2012), no. 4, 803867.