# American Institute of Mathematical Sciences

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

## On the injectivity radius in Hofer's geometry

 1 Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec, Canada 2 Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec, 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.
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, Ann. Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699. [2] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806. [3] J. Kędra and D. McDuff, Homotopy properties of Hamiltonian group actions, Geom. Topol., 9 (2005), 121-162. doi: 10.2140/gt.2005.9.121. [4] , M. Khanevsky and F. Zapolsky,, Private communication., (). [5] F. Lalonde and D. McDuff, Hofer's $L^{\infty}$-geometry: Energy and stability of Hamiltonian flows. Part II, Invent. Math., 122 (1995), 35-69. doi: 10.1007/BF01231438. [6] F. Lalonde and C. Pestieau, Stabilisation of symplectic inequalities and applications, in Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999, 63-71. [7] L. Polterovich, Hofer's diameter and Lagrangian intersections, Internat. Math. Res. Notices, 4 (1998), 217-223. doi: 10.1155/S1073792898000178. [8] Y. Savelyev, Virtual Morse theory on $\Omega\text{Ham}(M, \omega)$, J. Differ. Geom., 84 (2010), 409-425. [9] _____, Bott periodicity and stable quantum classes, Selecta Math. (N.S.), 19 (2013), 439-460. [10] _____, Quantum characteristic classes and the Hofer metric, Geom. Topol., 12 (2008), 2277-2326. [11] P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal., 7 (1997), 1046-1095. doi: 10.1007/s000390050037. [12] Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing Co., Singapore, 2001. doi: 10.1142/9789812811622.

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##### References:
 [1] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699. [2] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806. [3] J. Kędra and D. McDuff, Homotopy properties of Hamiltonian group actions, Geom. Topol., 9 (2005), 121-162. doi: 10.2140/gt.2005.9.121. [4] , M. Khanevsky and F. Zapolsky,, Private communication., (). [5] F. Lalonde and D. McDuff, Hofer's $L^{\infty}$-geometry: Energy and stability of Hamiltonian flows. Part II, Invent. Math., 122 (1995), 35-69. doi: 10.1007/BF01231438. [6] F. Lalonde and C. Pestieau, Stabilisation of symplectic inequalities and applications, in Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999, 63-71. [7] L. Polterovich, Hofer's diameter and Lagrangian intersections, Internat. Math. Res. Notices, 4 (1998), 217-223. doi: 10.1155/S1073792898000178. [8] Y. Savelyev, Virtual Morse theory on $\Omega\text{Ham}(M, \omega)$, J. Differ. Geom., 84 (2010), 409-425. [9] _____, Bott periodicity and stable quantum classes, Selecta Math. (N.S.), 19 (2013), 439-460. [10] _____, Quantum characteristic classes and the Hofer metric, Geom. Topol., 12 (2008), 2277-2326. [11] P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal., 7 (1997), 1046-1095. doi: 10.1007/s000390050037. [12] Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing Co., Singapore, 2001. doi: 10.1142/9789812811622.
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