On the Hofer Girth of the Sphere of Great Circles

Abstract

An oriented equator of $\mathbb{S}^2$ is the image of an oriented embedding $\mathbb{S}^1 \hookrightarrow \mathbb{S}^2$ such that it divides $\mathbb{S}^2$ into two equal area halves. Following Chekanov, we define the Hofer distance between two oriented equators as the infimal Hofer norm of a Hamiltonian diffeomorphism taking one to another. Consider $\mathcal{E}q_+$ the space of oriented equators. We define the Hofer girth of an embedding $j:\mathbb{S}^2 \hookrightarrow \mathcal{E}q_+$ as the infimum of the Hofer diameter of $j'(\mathbb{S}^2)$, where $j'$ is homotopic to $j$. There is a natural embedding $i_0:\mathbb{S}^2\hookrightarrow\mathcal{E}q_+$, sending a point on the sphere to the positively oriented great circle perpendicular to it. In this paper we provide an upper bound on the Hofer girth of $i_0$.