Lagrangian submanifolds in strict nearly Kähler 6-manifolds
Hông Vân Lê and Lorenz J. Schwachhöfer
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Submission date: 14. Sep. 2015
published in: Osaka journal of mathematics, 56 (2019) 3, p. 601-629
MSC-Numbers: 53C40, 53C38, 53D12, 58D99
Keywords and phrases: nearly Kähler 6-manifold, Lagrangian submanifold, calibration, Jacobi field, moduli space
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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.