On the injectivity radius in Hofer's geometry

François Lalonde; Yasha Savelyev

doi:10.3934/era.2014.21.177

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2014, Volume 21: 177-185. Doi: 10.3934/era.2014.21.177

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On the injectivity radius in Hofer's geometry

François Lalonde^1, and
Yasha Savelyev^2,

1.
Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec
2.
Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec

Received: March 31, 2014

Revised: July 31, 2014

Published: January 2014

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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 the weak form of the conjecture.

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Mathematics Subject Classification: 53C15, 53D12, 53D40, 53D45, 57R58, 57S05, 58B20.

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