On the injectivity radius in Hofer's geometry

Previous Article
Globally subanalytic CMC surfaces in $\mathbb{R}^3$
ERA-MS Home
This Volume
Next Article
Groups of Lie type, vertex algebras, and modular moonshine

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

Abstract
Related Papers

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, 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.

show all references

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.

[1]	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
[2]	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
[3]	Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109
[4]	Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811
[5]	Gennadi Sardanashvily. Lagrangian dynamics of submanifolds. Relativistic mechanics. Journal of Geometric Mechanics, 2012, 4 (1) : 99-110. doi: 10.3934/jgm.2012.4.99
[6]	Michael Khanevsky. Hofer's length spectrum of symplectic surfaces. Journal of Modern Dynamics, 2015, 9: 219-235. doi: 10.3934/jmd.2015.9.219
[7]	Sylvain E. Cappell, Anatoly Libgober, Laurentiu Maxim and Julius L. Shaneson. Hodge genera and characteristic classes of complex algebraic varieties. Electronic Research Announcements, 2008, 15: 1-7. doi: 10.3934/era.2008.15.1
[8]	Cédric M. Campos, Elisa Guzmán, Juan Carlos Marrero. Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (1) : 1-26. doi: 10.3934/jgm.2012.4.1
[9]	Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375
[10]	Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
[11]	Mathieu Molitor. On the relation between geometrical quantum mechanics and information geometry. Journal of Geometric Mechanics, 2015, 7 (2) : 169-202. doi: 10.3934/jgm.2015.7.169
[12]	Paolo Baroni, Agnese Di Castro, Giampiero Palatucci. Intrinsic geometry and De Giorgi classes for certain anisotropic problems. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 647-659. doi: 10.3934/dcdss.2017032
[13]	Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.
[14]	Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77
[15]	Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425
[16]	Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone. Schwinger's picture of quantum mechanics: 2-groupoids and symmetries. Journal of Geometric Mechanics, 2021, 13 (3) : 333-354. doi: 10.3934/jgm.2021008
[17]	Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[18]	Scott Crass. Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168. Journal of Modern Dynamics, 2007, 1 (2) : 175-203. doi: 10.3934/jmd.2007.1.175
[19]	Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157
[20]	Egil Bae, Xue-Cheng Tai, Wei Zhu. Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours. Inverse Problems and Imaging, 2017, 11 (1) : 1-23. doi: 10.3934/ipi.2017001

2020 Impact Factor: 0.929

Tools

Metrics

Other articles
by authors

on AIMS
François Lalonde Yasha Savelyev
on Google Scholar
François Lalonde Yasha Savelyev

[Back to Top]