Contact Homology, Capacity and Non-Squeezing in via Generating Functions
- [1] Instituto Superior Técnico Departamento de Matemática Av. Rovisco Pais 1049-001 Lisboa (Portugal)
Annales de l’institut Fourier (2011)
- Volume: 61, Issue: 1, page 145-185
- ISSN: 0373-0956
Access Full Article
topAbstract
topHow to cite
topSandon, Sheila. "Contact Homology, Capacity and Non-Squeezing in $\mathbb{R}^{2n}\times S^{1}$ via Generating Functions." Annales de l’institut Fourier 61.1 (2011): 145-185. <http://eudml.org/doc/219854>.
@article{Sandon2011,
abstract = {Starting from the work of Bhupal we extend to the contact case the Viterbo capacity and Traynor’s construction of symplectic homology. As an application we get a new proof of the Non-Squeezing Theorem of Eliashberg, Kim and Polterovich.},
affiliation = {Instituto Superior Técnico Departamento de Matemática Av. Rovisco Pais 1049-001 Lisboa (Portugal)},
author = {Sandon, Sheila},
journal = {Annales de l’institut Fourier},
keywords = {Contact non-squeezing; contact capacity; contact homology; orderability of contact manifolds; generating functions; contact non-squeezing},
language = {eng},
number = {1},
pages = {145-185},
publisher = {Association des Annales de l’institut Fourier},
title = {Contact Homology, Capacity and Non-Squeezing in $\mathbb\{R\}^\{2n\}\times S^\{1\}$ via Generating Functions},
url = {http://eudml.org/doc/219854},
volume = {61},
year = {2011},
}
TY - JOUR
AU - Sandon, Sheila
TI - Contact Homology, Capacity and Non-Squeezing in $\mathbb{R}^{2n}\times S^{1}$ via Generating Functions
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 1
SP - 145
EP - 185
AB - Starting from the work of Bhupal we extend to the contact case the Viterbo capacity and Traynor’s construction of symplectic homology. As an application we get a new proof of the Non-Squeezing Theorem of Eliashberg, Kim and Polterovich.
LA - eng
KW - Contact non-squeezing; contact capacity; contact homology; orderability of contact manifolds; generating functions; contact non-squeezing
UR - http://eudml.org/doc/219854
ER -
References
top- M. Bhupal, Legendrian intersections in the 1-jet bundle, (1998)
- M. Bhupal, A partial order on the group of contactomorphisms of via generating functions, Turkish J. Math. 25 (2001), 125-135 Zbl1007.53067MR1829083
- P. Biran, L. Polterovich, D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003), 65-118 Zbl1034.53089MR1991647
- M. Chaperon, Une idée du type “géodésiques brisées” pour les systémes hamiltoniens, C. R. Acad. Sci. Paris, Sér. I Math. 298 (1984), 293-296 Zbl0576.58010MR765426
- M. Chaperon, On generating families, The Floer Memorial Volume 133 (1995), 283-296, H. Hofer et al., Basel Zbl0837.58003MR1362831
- Y. Chekanov, Critical points of quasi-functions and generating families of Legendrian manifolds, Funct. Anal. Appl. 30 (1996), 118-128 Zbl0873.58017MR1402081
- Y. Chekanov, O. van Koert, F. Schlenk, Minimal atlases of closed contact manifolds, arXiv:0807.3047 Zbl1186.53096
- Y. Chekanov, P. Pushkar, Combinatorics of fronts of Legendrian links, and Arnold’s 4-conjectures, Russian Math. Surveys 60 (2005), 95-149 Zbl1085.57008MR2145660
- V. Chernov, S. Nemirovski, Legendrian links, causality, and the Low conjecture, arXiv:0810.5091v2 Zbl1186.83013
- V. Chernov, S. Nemirovski, Non-negative Legendrian isotopy in , arXiv:0905.0983 Zbl1194.53066
- K. Cieliebak, V. Ginzburg, E. Kerman, Symplectic homology and periodic orbits near symplectic submanifolds, Comment. Math. Helv. 79 (2004), 554-581 Zbl1073.53118MR2081726
- V. Colin, E. Ferrand, P. Pushkar, Positive loops of Legendrian embeddings, (2007)
- P. Eiseman, J. Lima, J. Sabloff, L. Traynor, A partial ordering on slices of planar Lagrangians, J. Fixed Point Theory Appl. 3 (2008), 431-447 Zbl1149.53318MR2434456
- Y. Eliashberg, New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc. 4 (1991), 513-520 Zbl0733.58011MR1102580
- Y. Eliashberg, M. Gromov, Lagrangian intersection theory : finite-dimensional approach, Geometry of differential equations 186 (1998), Amer. Math. Soc., Providence, RI Zbl0919.58015MR1732407
- Y. Eliashberg, S. S. Kim, L. Polterovich, Geometry of contact transformations and domains: orderability vs squeezing, Geom. and Topol. 10 (2006), 1635-1747 Zbl1134.53044MR2284048
- Y. Eliashberg, L. Polterovich, Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal. 10 (2000), 1448-1476 Zbl0986.53036MR1810748
- E. Ferrand, P. Pushkar, Morse theory and global coexistence of singularities on wave fronts, J. London Math. Soc. 74 (2006), 527-544 Zbl1113.57015MR2269593
- D. Fuchs, D. Rutherford, Generating families and Legendrian contact homology in the standard contact space, arXiv:0807.4277 Zbl1237.57026
- H. Geiges, An Introduction to Contact Topology, (2008), Cambridge University Press Zbl1153.53002MR2397738
- V. Ginzburg, B. Gürel, Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles, Duke Math. J. 123 (2004), 1-47 Zbl1066.53138MR2060021
- A. Givental, Nonlinear generalization of the Maslov index, Theory of singularities and its applications 1 (1990), 71-103, Amer. Math. Soc., Providence, RI Zbl0728.53024MR1089671
- A. Givental, A symplectic fixed point theorem for toric manifolds, The Floer Memorial Volume 133 (1995), 445-481, H. Hofer et al., Basel Zbl0835.55001MR1362837
- M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent.Math. 82 (1985), 307-347 Zbl0592.53025MR809718
- D. Hermann, Inner and outer Hamiltonian capacities, Bull. Soc. Math. France 132 (2004), 509-541 Zbl1083.53083MR2131902
- H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, (1994), Birkhäuser Zbl0805.58003MR1306732
- L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 17-183 Zbl0212.46601MR388463
- J. Jordan, L. Traynor, Generating family invariants for Legendrian links of unknots, Algebr. Geom. Topol. 6 (2006), 895-933 Zbl1130.57018MR2240920
- F. Laudenbach, J. C. Sikorav, Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibre cotangent, Invent. Math. 82 (1985), 349-357 Zbl0592.58023MR809719
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, (1998), Oxford University Press Zbl1066.53137MR1698616
- D. Milinković, Morse homology for generating functions of Lagrangian submanifolds, Trans. Amer. Math. Soc. 351 (1999), 3953-3974 Zbl0938.53043MR1475690
- J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale, C.R. Acad. Sci. Paris, Sér. I Math. 302 (1986), 119-122 Zbl0602.58019MR830282
- J. C. Sikorav, Problemes d’intersections et de points fixes en géométrie hamiltonienne, Comment. Math. Helv. 62 (1987), 62-73 Zbl0684.58015MR882965
- D. Théret, Utilisation des fonctions génératrices en géométrie symplectique globale, (1995)
- D. Théret, Rotation numbers of Hamiltonian isotopies in complex projective spaces, Duke Math. J. 94 (1998), 13-27 Zbl0976.53093MR1635892
- D. Théret, A complete proof of Viterbo’s uniqueness theorem on generating functions, Topology Appl. 96 (1999), 249-266 Zbl0952.53037MR1709692
- L. Traynor, Symplectic Homology via generating functions, Geom. Funct. Anal. 4 (1994), 718-748 Zbl0822.58020MR1302337
- L. Traynor, Generating Function Polynomials for Legendrian Links, Geom. and Topol. 5 (2001), 719-760 Zbl1030.53086MR1871403
- C. Viterbo, Functors and computations in Floer homology with applications, Part II Zbl0954.57015
- C. Viterbo, Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systémes hamiltoniens, Bull. Soc. Math. France 115 (1987), 361-390 Zbl0639.58018MR926533
- C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685-710 Zbl0735.58019MR1157321
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.