Hofer Geometry of a Subset of a Symplectic Manifold

TL;DR

This paper introduces a generalized Hofer norm for Hamiltonian diffeomorphisms associated with subsets of symplectic manifolds and investigates the relative Hofer diameter for specific subsets like spheres and compact sets.

Contribution

It defines a semi-norm on Hamiltonian diffeomorphisms for subsets of symplectic manifolds and analyzes the relative Hofer diameter for various geometric configurations.

Findings

The Hofer diameter of the unit sphere in R^{2n} is at least π/2 for n≥2.

Existence of compact sets in R^{2n} with controlled Hausdorff dimension and a lower bound on their relative Hofer diameter.

Explicit bounds involving a function k(n,d) for the Hofer diameter of certain subsets.

Abstract

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.