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

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.Comment: 47 page

Coisotropic Submanifolds, Leafwise Fixed Points, and Presymplectic Embeddings

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

The Invariant Symplectic Action and Decay for Vortices

The (local) invariant symplectic action functional \A is associated to a Hamiltonian action of a compact connected Lie group \G on a symplectic manifold $(M,\omega)$, endowed with a \G-invariant Riemannian metric $_M$. It is defined on the set of pairs of loops (x,\xi):S^1\to M\x\Lie\G for which $x$ satisfies some admissibility condition. I prove a sharp isoperimetric inequality for \A if $_M$ is induced by some $\omega$-compatible and \G-invariant almost complex structure $J$, and, as an application, an optimal result about the decay at $\infty$ of symplectic vortices on the half-cylinder [0,\infty)\x S^1.Comment: 20 pages. Final version. (I shortened the version # 3 a lot.) Accepted for publication by J. Symplectic Geo

Note on coisotropic Floer homology and leafwise fixed points

For an adiscal or monotone regular coisotropic submanifold $N$ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $N$. Given a Hamiltonian isotopy $\phi=(\phi^t)$ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $(N,\phi)$-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold. Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine. The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.Comment: 11 pages. I have split this article off from "Leafwise fixed points for $C^0$-small Hamiltonian flows" arXiv:1408.4578 . version 2: I included a definition of Floer homology for an adiscal or monotone regular coisotropic submanifol

Coisotropic Displacement and Small Subsets of a Symplectic Manifold

We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly squeezable" set in $\mathbb{R}^{2n}$ of Hausdorff dimension at most $d$, for every $n\geq2$ and $d\geq n$. 4. Existence of a stably exotic symplectic form on $\mathbb{R}^{2n}$, for every $n\geq2$. 5. Non-triviality of a new capacity, which is based on the minimal symplectic area of a regular coisotropic submanifold of dimension $d$.Comment: 34 page

A Symplectically Non-Squeezable Small Set and the Regular Coisotropic Capacity

We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $R^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the $d$-th regular coisotropic capacity, which is sharp up to a factor of 3. For an open subset of a geometrically bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by M. Audin and L. Polterovich.Comment: 15 pages, v2: added references to articles by H. Geiges and K. Zehmisch, v3: added "for $n\geq2$" in the abstract, v4: streamlined the introduction and simplified the proof of two-dimensional squeezing (Proposition 4