Cyclic Lagrangian Submanifolds and Lagrangian Fibrations

Abstract

Let $(M,\omega)$ be a symplectic manifold and $L\subset M$ be a Lagrangian submanifold. In [Oh2], the cyclic condition of $L$ was defined. Y.-G. Oh proved that, in [Oh2], if $(M,\omega)$ is Kähler-Einstein with non-zero scalar curvature and $L$ is minimal, then $L$ is cyclic. In this article, first, we prove that $L$ is cyclic if and only if the ``mean cuvature cohomology class'' of $L$ is rational, when $(M,\omega)$ is Kähler-Einstein with non-zero scalar curvature. Secondly, we see that there are non-cyclic minimal Lagrangian submanifolds when $(M,\omega)$ is a prequantizable Ricci-flat Kähler manifold. Thirdly, if $(M,\omega)$ is Kähler-Einstein with non-zero scalar curvature, there are not minimal Lagrangian fibration structures on $M$ by a result of [Oh2]. Nevertheless we construct Hamiltonian minimal Lagrangian fibration.