Osaka Journal of Mathematics

Lagrangian submanifolds in strict nearly Kähler 6-manifolds

Hông Vân Lê and Lorenz Schwachhöfer

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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 $(M, J, g)$ 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 $(M^6, J, g)$ 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 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 [34].

Article information

Source
Osaka J. Math., Volume 56, Number 3 (2019), 601-629.

Dates
First available in Project Euclid: 16 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1563242426

Mathematical Reviews number (MathSciNet)
MR3981303

Zentralblatt MATH identifier
07108032

Subjects
Primary: 53C40: Global submanifolds [See also 53B25] 53C38: Calibrations and calibrated geometries 53D12: Lagrangian submanifolds; Maslov index 58D99: None of the above, but in this section

Citation

Lê, Hông Vân; Schwachhöfer, Lorenz. Lagrangian submanifolds in strict nearly Kähler 6-manifolds. Osaka J. Math. 56 (2019), no. 3, 601--629. https://projecteuclid.org/euclid.ojm/1563242426


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Osaka University and Osaka City University, Departments of Mathematics

Osaka Journal of Mathematics

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