Submanifolds and the Hofer norm

Usher, Michael. "Submanifolds and the Hofer norm." Journal of the European Mathematical Society 016.8 (2014): 1571-1616. <http://eudml.org/doc/277803>.

@article{Usher2014,
abstract = {In [Ch00], Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we consider the orbits of more general submanifolds. We show that, for the Chekanov–Hofer pseudometric on the orbit of a closed submanifold to be a genuine metric, it is necessary for the submanifold to be coisotropic, and we show that this condition is sufficient under various additional geometric assumptions. At the other extreme, we show that the image of a generic closed embedding with any codimension larger than one is “weightless,” in the sense that the Chekanov–Hofer pseudometric on its orbit vanishes identically. In particular this yields examples of submanifolds which have zero displacement energy but are not infinitesimally displaceable.},
author = {Usher, Michael},
journal = {Journal of the European Mathematical Society},
keywords = {Hamiltonian diffeomorphism; Hofer metric; coisotropic submanifold; symplectic rigidity; weightlessness; Hamiltonian diffeomorphisms; Hofer metric; coisotropic submanifold; symplectic rigidity; weightlessness},
language = {eng},
number = {8},
pages = {1571-1616},
publisher = {European Mathematical Society Publishing House},
title = {Submanifolds and the Hofer norm},
url = {http://eudml.org/doc/277803},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Usher, Michael
TI - Submanifolds and the Hofer norm
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 8
SP - 1571
EP - 1616
AB - In [Ch00], Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we consider the orbits of more general submanifolds. We show that, for the Chekanov–Hofer pseudometric on the orbit of a closed submanifold to be a genuine metric, it is necessary for the submanifold to be coisotropic, and we show that this condition is sufficient under various additional geometric assumptions. At the other extreme, we show that the image of a generic closed embedding with any codimension larger than one is “weightless,” in the sense that the Chekanov–Hofer pseudometric on its orbit vanishes identically. In particular this yields examples of submanifolds which have zero displacement energy but are not infinitesimally displaceable.
LA - eng
KW - Hamiltonian diffeomorphism; Hofer metric; coisotropic submanifold; symplectic rigidity; weightlessness; Hamiltonian diffeomorphisms; Hofer metric; coisotropic submanifold; symplectic rigidity; weightlessness
UR - http://eudml.org/doc/277803
ER -