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In this work, we establish new rigidity results for the Maslov class ofLagrangian submanifolds in large classes of closed and convex symplecticmanifolds. Our main result establishes upper bounds for the minimal Maslovnumber of displaceable Lagrangian submanifolds which are product manifoldswhose factors each admit a metric of negative sectional curvature. SuchLagrangian submanifolds exist in every symplectic manifold of dimension greaterthan six or equal to four.  The proof utilizes the relations between closed geodesics on the Lagrangian,the periodic orbits of geometric Hamiltonian flows supported near theLagrangian, and the length minimizing properties of these flows with respect tothe negative Hofer length functional.

Maslov class rigidity for Lagrangian submanifolds via Hofer's geometry