How large is the shadow of a symplectic ball?
Abstract
Consider the image of a 2n-dimensional unit ball by an open symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection into a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromov's non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.
Keywords
Cite
@article{arxiv.1202.3614,
title = {How large is the shadow of a symplectic ball?},
author = {Alberto Abbondandolo and Rostislav Matveyev},
journal= {arXiv preprint arXiv:1202.3614},
year = {2013}
}
Comments
Final version, identical to the published one. Added comments about the relationship with a conjecture of Viterbo and related references. Some of the results of this paper are contained in our previous preprint arXiv:1105.2931, which is no longer updated and will never become a published article