On the Hofer Girth of the Sphere of Great Circles

TL;DR

This paper investigates the Hofer girth of a natural embedding of the sphere into the space of oriented equators, providing an upper bound for this geometric measure using concepts from symplectic topology.

Contribution

The paper introduces the concept of Hofer girth for embeddings into the space of oriented equators and establishes an upper bound for the specific embedding related to the sphere.

Findings

Upper bound on the Hofer girth of the natural embedding

Connection between Hofer distance and symplectic topology

Insights into the geometry of equators on the sphere

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 $i_0$.