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June 2005 Cyclic Lagrangian Submanifolds and Lagrangian Fibrations
Hajime ONO
Tokyo J. Math. 28(1): 63-70 (June 2005). DOI: 10.3836/tjm/1244208279

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.

Citation

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Hajime ONO. "Cyclic Lagrangian Submanifolds and Lagrangian Fibrations." Tokyo J. Math. 28 (1) 63 - 70, June 2005. https://doi.org/10.3836/tjm/1244208279

Information

Published: June 2005
First available in Project Euclid: 5 June 2009

zbMATH: 1089.53053
MathSciNet: MR2149623
Digital Object Identifier: 10.3836/tjm/1244208279

Rights: Copyright © 2005 Publication Committee for the Tokyo Journal of Mathematics

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Vol.28 • No. 1 • June 2005
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