Coisotropic Submanifolds, Leafwise Fixed Points, and Presymplectic Embeddings

Let $(M,\omega )$ be a geometrically bounded symplectic manifold, $N\subseteq M$aclosed,regular(i\mathrm{.}e\mathrm{.}\mathrm{"}fibering\mathrm{"})coisotropicsubmanifold,and$\phi:M\to M$ a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of $\varphi$ is bounded below by the sum of the $Z_2$-Betti numbers of $N$, provided that the Hofer distance between $\varphi$ and the identity is small enough and the pair $(N,\varphi )$ 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.