Hofer's metrics and boundary depth
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 1, p. 57-129

We show that if (M,ω) 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,ω) 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 M×M 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.

Nous montrons que si (M,ω) 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,ω) 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 M×MM 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.

DOI : https://doi.org/10.24033/asens.2185
Classification:  53D22,  53D40
Keywords: Hofer metric, hamiltonian diffeomorphism, lagrangian submanifold, Floer complex
@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {1},
     year = {2013},
     pages = {57-129},
     doi = {10.24033/asens.2185},
     zbl = {1271.53076},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2013_4_46_1_57_0}
}
Usher, Michael. Hofer's metrics and boundary depth. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 1, pp. 57-129. doi : 10.24033/asens.2185. http://www.numdam.org/item/ASENS_2013_4_46_1_57_0/

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