How large is the shadow of a symplectic ball?

Details How large is the shadow of a symplectic ball? How large is the shadow of a symplectic ball?

2012-02-16

arxiv.org/abs/1202.3614v1

Mathematics

Symplectic Geometry

This paper collects the results of our previous preprint "Middle-dimensional squeezing and non-squeezing behavior of symplec

Scientific paper

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.

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