On weakly exact Lagrangians in Liouville bi-fillings

Here we study several questions concerning Liouville domains that are diffeomorphic to cylinders, so called trivial bi-fillings, for which the Liouville skeleton moreover is smooth and of codimension one; we also propose the notion of a Liouville-Hamiltonian structure, which encodes the symplectic structure of a hypersurface tangent to the Liouville flow, e.g. the skeleta of certain bi-fillings. We show that the symplectic homology of a bi-filling is non-trivial, and that a connected Lagrangian inside a bi-filling whose boundary lives in different components of the contact boundary automatically has non-vanishing wrapped Floer cohomology. We also prove geometric vanishing and non-vanishing criteria for the wrapped Floer cohomology of an exact Lagrangian with disconnected cylindrical ends. Finally, we give homotopy-theoretic restrictions on the closed weakly exact Lagrangians in the McDuff and torus bundle Liouville domains.