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 $\varphi :M\to M$ 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.

41 pages. I added a discussion about optimality of the bounds on the number of leafwise fixed points and on the Hofer distance

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• A Maslov Map for Coisotropic Submanifolds, Leaf-wise Fixed Points and Presymplectic Non-Embeddings
• Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds
• Künneth Formula in Rabinowitz Floer homology
• On the Maslov class rigidity for coisotropic submanifolds