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

Abstract

Let $(M,\omega )$ be a symplectic manifold, $N\subseteq M$ a coisotropic submanifold, and $\Sigma$ a compact oriented (real) surface. I define a natural Maslov index for each continuous map $u:\Sigma \to M$ that sends every connected component of $\partial \Sigma$ 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 $N<\dim M/2$. 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 $(M,\omega )$ 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.

47 pages

References in corpus (2)

• Geometry of coisotropic submanifolds in symplectic and Kähler manifolds
• On the Maslov class rigidity for coisotropic submanifolds

Cited by in corpus (1)

• On the Maslov class rigidity for coisotropic submanifolds