A Maslov Map for Coisotropic Submanifolds, Leaf-wise Fixed Points and Presymplectic Non-Embeddings

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Let (M,ω) be a symplectic manifold, N M a coisotropic submanifold, and Σ a compact oriented (real) surface. I define a natural Maslov index for each continuous map u:Σ→ M that sends every connected component of ∂Σ to some isotropic leaf of N. This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of N to compensate for the loss of boundary data in the case codimension N< M/2. The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplectic automorphisms. I prove a lower bound on the number of leafwise fixed points of a Hamiltonian diffeomorphism, if (M,ω) is geometrically bounded and N is closed, regular (i.e. "fibering"), and monotone. As an application, we obtain a presymplectic non-embedding result. I also prove a coisotropic version of the Audin conjecture.