Calabi quasimorphisms for the symplectic ball
Abstract
We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show that the (Lagrangian) Clifford torus in the complex projective space cannot be displaced from itself by a Hamiltonian isotopy.
Cite
@article{arxiv.math/0307011,
title = {Calabi quasimorphisms for the symplectic ball},
author = {Paul Biran and Michael Entov and Leonid Polterovich},
journal= {arXiv preprint arXiv:math/0307011},
year = {2007}
}
Comments
Minor errors corrected. To appear in Communications in Contemporary Mathematics