Hyperbolic 3-manifolds admitting no fillable contact structures

Given an irreducible 3-manifold M M , Eliashberg asked whether M M admits a tight or fillable contact structure. When M M is a Seifert fibered space, this question is completely solved. However, it has not been answered for any hyperbolic 3-manifold M M , even for the existence question of fillable contact structures. In this paper, we find infinitely many hyperbolic 3-manifolds that admit no weakly symplectically fillable contact structures, using tools in Heegaard Floer theory. We also remark that some of these manifolds do admit tight contact structures.