Hofer’s metrics and boundary depth

Michael Usher

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 1, page 57-129
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Usher, Michael. "Hofer’s metrics and boundary depth." Annales scientifiques de l'École Normale Supérieure 46.1 (2013): 57-129. <http://eudml.org/doc/272152>.

@article{Usher2013,
abstract = {We show that if $(M,\omega )$ 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 $(M,\omega )$ 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 $M\times M$ 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.},
author = {Usher, Michael},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Hofer metric; hamiltonian diffeomorphism; lagrangian submanifold; Floer complex},
language = {eng},
number = {1},
pages = {57-129},
publisher = {Société mathématique de France},
title = {Hofer’s metrics and boundary depth},
url = {http://eudml.org/doc/272152},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Usher, Michael
TI - Hofer’s metrics and boundary depth
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 1
SP - 57
EP - 129
AB - We show that if $(M,\omega )$ 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 $(M,\omega )$ 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 $M\times M$ 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.
LA - eng
KW - Hofer metric; hamiltonian diffeomorphism; lagrangian submanifold; Floer complex
UR - http://eudml.org/doc/272152
ER -

References

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