\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the injectivity radius in Hofer's geometry

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: 53C15, 53D12, 53D40, 53D45, 57R58, 57S05, 58B20.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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]
    [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(75) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return