In McDuff and torus bundle Liouville domains, every weakly exact Lagrangian torus is homotopic to a standard fibre, and any exact Lagrangian meeting two different boundary components has non-zero wrapped Floer cohomology.
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On weakly exact Lagrangians in Liouville bi-fillings
In McDuff and torus bundle Liouville domains, every weakly exact Lagrangian torus is homotopic to a standard fibre, and any exact Lagrangian meeting two different boundary components has non-zero wrapped Floer cohomology.