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

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, Canada
2. 

Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec, Canada

Received  April 2014 Revised  August 2014 Published  December 2014

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.
Keywords: Lagrangian submanifolds, Hofer's geometry, Quantum characteristic classes..
Mathematics Subject Classification: 53C15, 53D12, 53D40, 53D45, 57R58, 57S05, 58B2.
Citation: François Lalonde, Yasha Savelyev. On the injectivity radius in Hofer's geometry. Electronic Research Announcements, 2014, 21: 177-185. doi: 10.3934/era.2014.21.177
References:
[1]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, \emph{Ann. Math. (2), 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[2]

M. Gromov, Pseudo holomorphic curves in symplectic manifolds,, \emph{Invent. Math.}, 82 (1985), 307.  doi: 10.1007/BF01388806.  Google Scholar

[3]

J. Kędra and D. McDuff, Homotopy properties of Hamiltonian group actions,, \emph{Geom. Topol.}, 9 (2005), 121.  doi: 10.2140/gt.2005.9.121.  Google Scholar

[4]

, M. Khanevsky and F. Zapolsky,, Private communication., ().   Google Scholar

[5]

F. Lalonde and D. McDuff, Hofer's $L^{\infty}$-geometry: Energy and stability of Hamiltonian flows. Part II,, \emph{Invent. Math.}, 122 (1995), 35.  doi: 10.1007/BF01231438.  Google Scholar

[6]

F. Lalonde and C. Pestieau, Stabilisation of symplectic inequalities and applications,, in \emph{Northern California Symplectic Geometry Seminar}, (1999), 63.   Google Scholar

[7]

L. Polterovich, Hofer's diameter and Lagrangian intersections,, \emph{Internat. Math. Res. Notices}, 4 (1998), 217.  doi: 10.1155/S1073792898000178.  Google Scholar

[8]

Y. Savelyev, Virtual Morse theory on $\Omega\text{Ham}(M, \omega)$,, \emph{J. Differ. Geom.}, 84 (2010), 409.   Google Scholar

[9]

_____, Bott periodicity and stable quantum classes,, \emph{Selecta Math. (N.S.)}, 19 (2013), 439.   Google Scholar

[10]

_____, Quantum characteristic classes and the Hofer metric,, \emph{Geom. Topol.}, 12 (2008), 2277.   Google Scholar

[11]

P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings,, \emph{Geom. Funct. Anal.}, 7 (1997), 1046.  doi: 10.1007/s000390050037.  Google Scholar

[12]

Z. Shen, Lectures on Finsler Geometry,, World Scientific Publishing Co., (2001).  doi: 10.1142/9789812811622.  Google Scholar

show all references

References:
[1]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, \emph{Ann. Math. (2), 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[2]

M. Gromov, Pseudo holomorphic curves in symplectic manifolds,, \emph{Invent. Math.}, 82 (1985), 307.  doi: 10.1007/BF01388806.  Google Scholar

[3]

J. Kędra and D. McDuff, Homotopy properties of Hamiltonian group actions,, \emph{Geom. Topol.}, 9 (2005), 121.  doi: 10.2140/gt.2005.9.121.  Google Scholar

[4]

, M. Khanevsky and F. Zapolsky,, Private communication., ().   Google Scholar

[5]

F. Lalonde and D. McDuff, Hofer's $L^{\infty}$-geometry: Energy and stability of Hamiltonian flows. Part II,, \emph{Invent. Math.}, 122 (1995), 35.  doi: 10.1007/BF01231438.  Google Scholar

[6]

F. Lalonde and C. Pestieau, Stabilisation of symplectic inequalities and applications,, in \emph{Northern California Symplectic Geometry Seminar}, (1999), 63.   Google Scholar

[7]

L. Polterovich, Hofer's diameter and Lagrangian intersections,, \emph{Internat. Math. Res. Notices}, 4 (1998), 217.  doi: 10.1155/S1073792898000178.  Google Scholar

[8]

Y. Savelyev, Virtual Morse theory on $\Omega\text{Ham}(M, \omega)$,, \emph{J. Differ. Geom.}, 84 (2010), 409.   Google Scholar

[9]

_____, Bott periodicity and stable quantum classes,, \emph{Selecta Math. (N.S.)}, 19 (2013), 439.   Google Scholar

[10]

_____, Quantum characteristic classes and the Hofer metric,, \emph{Geom. Topol.}, 12 (2008), 2277.   Google Scholar

[11]

P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings,, \emph{Geom. Funct. Anal.}, 7 (1997), 1046.  doi: 10.1007/s000390050037.  Google Scholar

[12]

Z. Shen, Lectures on Finsler Geometry,, World Scientific Publishing Co., (2001).  doi: 10.1142/9789812811622.  Google Scholar

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