On displaceability of pre-Lagrangian toric fibers in contact toric manifolds

In this paper, we analyze displaceability of pre-Lagrangian toric fibers in contact toric manifolds. While every symplectic toric manifold contains at least one non-displaceable Lagrangian toric fiber and infinitely many displaceable ones, we show that this is not the case for contact toric manifolds. More precisely, we prove that for the contact toric manifolds 𝕊2d−1(d ≥ 2) and 𝕋k × 𝕊2d+k−1(d ≥ 1) all pre-Lagrangian toric fibers are displaceable, and that for all contact toric manifolds for which the toric action is free, except possibly non-trivial 𝕋3-bundles over 𝕊2, all pre-Lagrangian toric fibers are non-displaceable. Moreover, we also prove that if for a compact connected contact toric manifold all but finitely many pre-Lagrangian toric fibers are non-displaceable then the action is necessarily free. On the other hand, as we will discuss, displaceability of all pre-Lagrangian toric fibers seems to be related to the non-orderability of the underlying contact manifolds.