[Métriques de Hofer et profondeur de bord]
Nous montrons que si 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 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 où 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.
We show that if 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 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 when 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
Mot clés : métrique de Hofer, difféomorphisme hamiltonien, sous-variété lagrangienne, complexe de Floer
@article{ASENS_2013_4_46_1_57_0, author = {Usher, Michael}, title = {Hofer's metrics and boundary depth}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {57--129}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 46}, number = {1}, year = {2013}, doi = {10.24033/asens.2185}, zbl = {1271.53076}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2185/} }
TY - JOUR AU - Usher, Michael TI - Hofer's metrics and boundary depth JO - Annales scientifiques de l'École Normale Supérieure PY - 2013 SP - 57 EP - 129 VL - 46 IS - 1 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2185/ DO - 10.24033/asens.2185 LA - en ID - ASENS_2013_4_46_1_57_0 ER -
%0 Journal Article %A Usher, Michael %T Hofer's metrics and boundary depth %J Annales scientifiques de l'École Normale Supérieure %D 2013 %P 57-129 %V 46 %N 1 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2185/ %R 10.24033/asens.2185 %G en %F ASENS_2013_4_46_1_57_0
Usher, Michael. Hofer's metrics and boundary depth. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 1, pp. 57-129. doi : 10.24033/asens.2185. http://www.numdam.org/articles/10.24033/asens.2185/
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