Unboundedness of the Lagrangian Hofer distance in the Euclidean ball

Previous Article
Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets
ERA-MS Home
This Volume
Next Article

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

Abstract
Related Papers

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.
[2]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676. doi: 10.1155/S1073792803210011.
[3]	M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[4]	M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.
[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.
[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.
[7]	M. Khanevsky, Hofer's metric on the space of diameters, J. Topol. Anal., 1 (2009), 407-416. doi: 10.1142/S1793525309000187.
[8]	R. Leclercq and F. Zapolsky, Spectral invariants for monotone Lagrangian submanifolds, in preparation.
[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.
[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.
[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.
[12]	M. Usher, Submanifolds and the Hofer norm, to appear in J. Eur. Math. Soc., arXiv:1201.2926.
[13]	C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.
[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.

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.
[2]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676. doi: 10.1155/S1073792803210011.
[3]	M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[4]	M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.
[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.
[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.
[7]	M. Khanevsky, Hofer's metric on the space of diameters, J. Topol. Anal., 1 (2009), 407-416. doi: 10.1142/S1793525309000187.
[8]	R. Leclercq and F. Zapolsky, Spectral invariants for monotone Lagrangian submanifolds, in preparation.
[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.
[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.
[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.
[12]	M. Usher, Submanifolds and the Hofer norm, to appear in J. Eur. Math. Soc., arXiv:1201.2926.
[13]	C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.
[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.

[1]	Michael Khanevsky. Hofer's length spectrum of symplectic surfaces. Journal of Modern Dynamics, 2015, 9: 219-235. doi: 10.3934/jmd.2015.9.219
[2]	Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205
[3]	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
[4]	José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921
[5]	Ely Kerman. Displacement energy of coisotropic submanifolds and Hofer's geometry. Journal of Modern Dynamics, 2008, 2 (3) : 471-497. doi: 10.3934/jmd.2008.2.471
[6]	Andrew James Bruce, Janusz Grabowski. Symplectic $ {\mathbb Z}_2^n $-manifolds. Journal of Geometric Mechanics, 2021, 13 (3) : 285-311. doi: 10.3934/jgm.2021020
[7]	Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
[8]	Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems and Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135
[9]	Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431
[10]	Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[11]	Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115
[12]	Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
[13]	Simon Hochgerner, Luis García-Naranjo. $G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball. Journal of Geometric Mechanics, 2009, 1 (1) : 35-53. doi: 10.3934/jgm.2009.1.35
[14]	Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.
[15]	B. Campos, P. Vindel. Transversal intersections of invariant manifolds of NMS flows on $S^{3}$. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 41-56. doi: 10.3934/dcds.2012.32.41
[16]	Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021, 13 (2) : 167-193. doi: 10.3934/jgm.2021002
[17]	Kun Shi, Guangcun Lu. Higher P-symmetric Ekeland-Hofer capacities. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1049-1070. doi: 10.3934/cpaa.2022009
[18]	Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
[19]	Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[20]	Chungen Liu, Qi Wang. Symmetrical symplectic capacity with applications. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2253-2270. doi: 10.3934/dcds.2012.32.2253

2020 Impact Factor: 0.929

Tools

Metrics

Other articles
by authors

on AIMS
Sobhan Seyfaddini
on Google Scholar
Sobhan Seyfaddini

[Back to Top]