  In this work, we establish new rigidity results for the Maslov class of
Lagrangian submanifolds in large classes of closed and convex symplectic
manifolds. Our main result establishes upper bounds for the minimal Maslov
number of displaceable Lagrangian submanifolds which are product manifolds
whose factors each admit a metric of negative sectional curvature. Such
Lagrangian submanifolds exist in every symplectic manifold of dimension greater
than 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 the
Lagrangian, and the length minimizing properties of these flows with respect to
the negative Hofer length functional.
