arXiv subjects

Fabian Ziltener

Publications and source records attributed to Fabian Ziltener.

22 records · Page 2Linked to original sources

arXiv2011

Hofer Geometry of a Subset of a Symplectic Manifold

To every closed subset $X$ of a symplectic manifold $(M,ω)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,ω)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,ω}$, generalizing the Hofer norm. We discuss $Ham(X,ω)$ and $\Vert\cdot\Vert^{X,ω}$ if $X$ is a symplectic or isotropic submanifold. The main result involves the relative Hofer diameter of $X$ in $M$. Its first part states that for the unit sphere in $R^{2n}$ this diameter is bounded below by $\frac\pi2$, if $n\geq2$. Its second part states that for $n\geq2$ and $d\geq n+1$ there exists a compact set in $R^{2n}$ of Hausdorff dimension at most $d$, with relative Hofer diameter bounded below by $π/k(n,d)$, where $k(n,d)$ is an explicitly defined integer.

math.SG↗

arXiv2009

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

Let $(M,ω)$ be a symplectic manifold, $N\subseteq M$ a coisotropic submanifold, and $Σ$ a compact oriented (real) surface. I define a natural Maslov index for each continuous map $u:Σ\to M$ that sends every connected component of $\partialΣ$ 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 $(M,ω)$ 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.

math.SG↗

arXiv2009

Coisotropic Submanifolds, Leafwise Fixed Points, and Presymplectic Embeddings

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

math.SG↗

arXiv2008

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 $(M,ω)$, endowed with a $\G$-invariant Riemannian metric $<\cdot,\cdot>_M$. It is defined on the set of pairs of loops $(x,ξ):S^1\to M\x\Lie\G$ for which $x$ satisfies some admissibility condition. I prove a sharp isoperimetric inequality for $\A$ if $<\cdot,\cdot>_M$ is induced by some $ω$-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 $[0,\infty)\x S^1$.

math.SG↗