紀要論文

紀要論文

text

Osaka Journal of Mathematics. 2019, 56(3), p. 601-629

LAGRANGIAN SUBMANIFOLDS IN STRICT NEARLY KӒHLER 6-MANIFOLDS

Osaka University and Osaka City University, Departments of Mathematics

英語

text

2019-07

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 𝐿³ 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].

10.18910/73359

73359

LAGRANGIAN SUBMANIFOLDS IN STRICT NEARLY KӒHLER 6-MANIFOLDS

Osaka University and Osaka City University, Departments of Mathematics

英語

2019-07

Osaka Journal of Mathematics

56

3

601

629

https://hdl.handle.net/11094/73359

10.18910/73359

https://doi.org/10.18910/73359