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Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 from minimal surfaces in 𝕊3

Part of: Global differential geometry Symplectic geometry, contact geometry Local differential geometry

Published online by Cambridge University Press:  27 December 2018

Burcu Bektaş ,
Marilena Moruz ,
Joeri Van der Veken  and
Luc Vrancken
Burcu Bektaş
Affiliation:
Faculty of Science and Letters, Department of Mathematics, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey (bektasbu@itu.edu.tr)
Marilena Moruz
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B - Box 2400, BE-3001 Leuven, Belgium (marilena.moruz@kuleuven.be)
Joeri Van der Veken
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B – Box 2400, BE-3001 Leuven, Belgium (joeri.vanderveken@wis.kuleuven.be)
Luc Vrancken
Affiliation:
LAMAV, ISTV2 Université de Valenciennes Campus du Mont Houy 59313 Valenciennes Cedex 9, France and KU Leuven Department of Mathematics, Celestijnenlaan 200B – Box 2400 BE-3001 Leuven, Belgium (luc.vrancken@univ-valenciennes.fr)
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Abstract

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

Keywords

Lagrangian submanifoldsNearly Kaehler 𝕊3 × 𝕊3minimal surfaces, 𝕊3

MSC classification

Primary: 53B25: Local submanifolds
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53D12: Lagrangian submanifolds; Maslov index
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 655 - 689
Copyright
Copyright © Royal Society of Edinburgh 2018 

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