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$\mathbb {\mathcal {C}}^{0}$-rigidity of Lagrangian submanifolds and punctured holomorphic disks in the cotangent bundle

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  03 November 2021

Cedric Membrez  and
Emmanuel Opshtein
Cedric Membrez
Affiliation:
UBS Emerging Technology Research, Zurich, Switzerlandckmembrez@gmail.com
Emmanuel Opshtein
Affiliation:
UFR de mathématiques, Université de Strasbourg, 7 rue René Descartes, 67084Strasbourg, Franceopshtein@unistra.fr
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Abstract

Our main result is the $\mathbb {\mathcal {C}}^{0}$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

Keywords

$\mathbb {\mathcal {C}}^{0}$-symplectic geometryLagrangian rigidity

MSC classification

Primary: 53D05: Symplectic manifolds, general 53D12: Lagrangian submanifolds; Maslov index
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 11 , November 2021 , pp. 2433 - 2493
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

Cedric Membrez has been partially supported by Swiss National Science Foundation grant 155540 and European Research Council Advanced grant 338809.

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