On the injectivity radius in Hofer's geometry

TL;DR

This paper investigates the local topology of the group of Hamiltonian diffeomorphisms under Hofer's metric, providing evidence supporting the conjecture that small metric balls are contractible.

Contribution

The authors prove several results that support the weak form of the conjecture regarding contractibility of small balls in Hofer's geometry.

Findings

Support for the weak conjecture on contractibility of small balls

Results indicating local contractibility in Hofer's geometry

Progress towards understanding the injectivity radius in symplectic topology

Abstract

In this note we consider the following conjecture: given any closed symplectic manifold $M$, there is a sufficiently small real positive number $\rho$ such that the open ball of radius $\rho$ in the Hofer metric centered at the identity on the group of Hamiltonian diffeomorphisms of $M$ is contractible, where the retraction takes place in that ball -- this is the strong version of the conjecture -- or inside the ambient group of Hamiltonian diffeomorphisms of $M$ -- this is the weak version of the conjecture. We prove several results that support that weak form of the conjecture.