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  2014, 21: 1-7. doi: 10.3934/era.2014.21.1

Unboundedness of the Lagrangian Hofer distance in the Euclidean ball

Sobhan Seyfaddini 1,

1. 

Département de Mathématiques et Applications de l'École Normale Supérieure, 45 rue d'Ulm, F 75230 Paris cedex 05, France

Received  October 2013 Revised  November 2013 Published  January 2014

Let $\mathcal{L}$ denote the space of Lagrangians Hamiltonian isotopic to the standard Lagrangian in the unit ball in $\mathbb{R}^{2n}$. We prove that the Lagrangian Hofer distance on $\mathcal{L}$ is unbounded.
Keywords: Hofer's distance, Symplectic manifolds, quasimorphisms..
Mathematics Subject Classification: Primary: 53D40; Secondary 37J0.
Citation: Sobhan Seyfaddini. Unboundedness of the Lagrangian Hofer distance in the Euclidean ball. Electronic Research Announcements, 2014, 21: 1-7. doi: 10.3934/era.2014.21.1
References:
[1]

P. Biran, M. Entov and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math., 6 (2004), 793-802. doi: 10.1142/S0219199704001525.  Google Scholar

[2]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676. doi: 10.1155/S1073792803210011.  Google Scholar

[3]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.  Google Scholar

[4]

M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.  Google Scholar

[5]

M. Entov, L. Polterovich and P. Py, On continuity of quasimorphisms for symplectic maps, in Perspectives in Analysis, Geometry, and Topology, Progr. Math., 296, Birkhäuser/Springer, New York, 2012, 169-197. doi: 10.1007/978-0-8176-8277-4_8.  Google Scholar

[6]

V. Humilière, Hofer's distance on diameters and the Maslov index, Int. Math. Res. Not. IMRN, 2012 (2012), 3415-3433. doi: 10.1093/imrn/rnr150.  Google Scholar

[7]

M. Khanevsky, Hofer's metric on the space of diameters, J. Topol. Anal., 1 (2009), 407-416. doi: 10.1142/S1793525309000187.  Google Scholar

[8]

R. Leclercq and F. Zapolsky, Spectral invariants for monotone Lagrangian submanifolds,, in preparation., ().   Google Scholar

[9]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in The breadth of symplectic and Poisson geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 525-570. doi: 10.1007/0-8176-4419-9_18.  Google Scholar

[10]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.  Google Scholar

[11]

S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds, J. Topol. Anal., 4 (2012), 481-498. doi: 10.1142/S1793525312500215.  Google Scholar

[12]

M. Usher, Submanifolds and the Hofer norm,, to appear in J. Eur. Math. Soc., ().   Google Scholar

[13]

C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.  Google Scholar

[14]

F. Zapolsky, On the Hofer geometry for weakly exact Lagrangian submanifolds, J. Symplectic Geom., 11 (2013), 475-488. doi: 10.4310/JSG.2013.v11.n3.a7.  Google Scholar

show all references

References:
[1]

P. Biran, M. Entov and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math., 6 (2004), 793-802. doi: 10.1142/S0219199704001525.  Google Scholar

[2]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676. doi: 10.1155/S1073792803210011.  Google Scholar

[3]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.  Google Scholar

[4]

M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.  Google Scholar

[5]

M. Entov, L. Polterovich and P. Py, On continuity of quasimorphisms for symplectic maps, in Perspectives in Analysis, Geometry, and Topology, Progr. Math., 296, Birkhäuser/Springer, New York, 2012, 169-197. doi: 10.1007/978-0-8176-8277-4_8.  Google Scholar

[6]

V. Humilière, Hofer's distance on diameters and the Maslov index, Int. Math. Res. Not. IMRN, 2012 (2012), 3415-3433. doi: 10.1093/imrn/rnr150.  Google Scholar

[7]

M. Khanevsky, Hofer's metric on the space of diameters, J. Topol. Anal., 1 (2009), 407-416. doi: 10.1142/S1793525309000187.  Google Scholar

[8]

R. Leclercq and F. Zapolsky, Spectral invariants for monotone Lagrangian submanifolds,, in preparation., ().   Google Scholar

[9]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in The breadth of symplectic and Poisson geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 525-570. doi: 10.1007/0-8176-4419-9_18.  Google Scholar

[10]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.  Google Scholar

[11]

S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds, J. Topol. Anal., 4 (2012), 481-498. doi: 10.1142/S1793525312500215.  Google Scholar

[12]

M. Usher, Submanifolds and the Hofer norm,, to appear in J. Eur. Math. Soc., ().   Google Scholar

[13]

C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.  Google Scholar

[14]

F. Zapolsky, On the Hofer geometry for weakly exact Lagrangian submanifolds, J. Symplectic Geom., 11 (2013), 475-488. doi: 10.4310/JSG.2013.v11.n3.a7.  Google Scholar

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