Hofer Geometry of a Subset of a Symplectic Manifold

Details Hofer Geometry of a Subset of a Symplectic Manifold Hofer Geometry of a Subset of a Symplectic Manifold

2011-02-24

arxiv.org/abs/1102.4889v1

Mathematics

Symplectic Geometry

34 pages

Scientific paper

To every closed subset $X$ of a symplectic manifold $(M,\omega)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,\omega)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,\omega}$, generalizing the Hofer norm. We discuss $Ham(X,\omega)$ and $\Vert\cdot\Vert^{X,\omega}$ 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 $\pi/k(n,d)$, where $k(n,d)$ is an explicitly defined integer.

Affiliated with

Also associated with

No associations

Rating

Hofer Geometry of a Subset of a Symplectic Manifold does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hofer Geometry of a Subset of a Symplectic Manifold, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hofer Geometry of a Subset of a Symplectic Manifold will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-86100