Semi-Local Exotic Lagrangian Tori in Dimension Four

We study exotic Lagrangian tori in dimension four. In certain Stein domains B₃₏ₐ (which naturally appear in almost toric fibrations) we find d+1 families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of B₃₏ₐ into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when X is compact and (almost) toric and hence conclude that X contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of X. In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces. Neither the embedded tori, nor the ambient space X needs to be monotone for our methods to work.