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Annales scientifiques de l'ENS - Titles - série 4, 46 (2013)

Titles < série 4, 46

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 46, fascicule 1 (2013)

Michael Usher
Hofer's metrics and boundary depth
Annales scientifiques de l'ENS 46, fascicule 1 (2013), 57-128

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Résumé :
Métriques de Hofer et profondeur de bord
Nous montrons que si (M,) est une variété symplectique fermée qui admet un champ vectoriel hamiltonien non-trivial dont toutes les orbites fermées contractiles sont constantes, la métrique de Hofer sur le groupe des difféomorphismes hamiltoniens de (M,) a alors un diamètre infini et admet donc des espaces vectoriels normés plongés quasi-isométriquement et de dimension infinie. Une conclusion semblable s'applique à la métrique de Hofer sur différents espaces de sous-variétés lagrangiennes, y compris les sous-variétés hamiltoniennes isotopiques à la diagonale en MMM satisfait à la condition dynamique ci-dessus. Pour prouver cela, nous utilisons les propriétés d'une quantité Floer-théorique appelée profondeur de bord, qui mesure la non-trivialité de l'opérateur limite sur le complexe de Floer de manière à encoder des informations robustes de topologie symplectique.

Mots-clefs : Métrique de Hofer, difféomorphisme hamiltonien, sous-variété lagrangienne, complexe de Floer

Abstract:
We show that if (M,) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of (M,) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer's metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in MM when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.

Keywords: Hofer metric, Hamiltonian diffeomorphism, Lagrangian submanifold, Floer complex

Class. math. : 53D22, 53D40


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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