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Lagrangian foliations and Anosov symplectomorphisms on Kähler manifolds

Part of: Symplectic geometry, contact geometry Dynamical systems with hyperbolic behavior Complex manifolds Global differential geometry Local differential geometry

Published online by Cambridge University Press:  30 October 2020

M. J. D. HAMILTON  and
D. KOTSCHICK Open the ORCID record for D. KOTSCHICK [Opens in a new window]
M. J. D. HAMILTON
Affiliation:
Institut für Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, 70569Stuttgart, Germany (e-mail: [email protected])
D. KOTSCHICK*
Affiliation:
Mathematisches Institut, LMU München, Theresienstr. 39, 80333München, Germany
*
e-mail: [email protected]
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Abstract

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We investigate parallel Lagrangian foliations on Kähler manifolds. On the one hand, we show that a Kähler metric admitting a parallel Lagrangian foliation must be flat. On the other hand, we give many examples of parallel Lagrangian foliations on closed flat Kähler manifolds which are not tori. These examples arise from Anosov automorphisms preserving a Kähler form.

Keywords

Lagrangian foliationflat Kähler manifoldAnosov symplectomorphism

MSC classification

Primary: 32Q15: Kähler manifolds 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 53B35: Hermitian and Kählerian structures
Secondary: 53C12: Foliations (differential geometric aspects) 53C55: Hermitian and Kählerian manifolds 53D12: Lagrangian submanifolds; Maslov index
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3325 - 3335
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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