Coisotropic Submanifolds, Leafwise Fixed Points, and Presymplectic Embeddings

Abstract

Let $(M,\omega)$ be a geometrically bounded symplectic manifold, $N\subseteq M$ a closed, regular (i.e. "fibering") coisotropic submanifold, and $\phi:M\to M$ a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of $\phi$ is bounded below by the sum of the $Z_2$-Betti numbers of $N$, provided that the Hofer distance between $\phi$ and the identity is small enough and the pair $(N,\phi)$ is non-degenerate. The bound is optimal if there exists a $Z_2$-perfect Morse function on $N$. A version of the Arnol'd-Givental conjecture for coisotropic submanifolds is also discussed. As an application, I prove a presymplectic non-embedding result.