Do uniruled six-manifolds contain Sol Lagrangian submanifolds?

Abstract

We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Koll\'ar. Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.