Recent advances in symplectic flexibility

Eliashberg, Yakov

doi:10.1090/S0273-0979-2014-01470-3

Recent advances in symplectic flexibility
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Bull. Amer. Math. Soc. 52 (2015), 1-26 Request permission

Abstract:

Flexible and rigid methods coexisted in symplectic topology from its inception. While the rigid methods dominated the development of the subject during the last three decades, the balance has somewhat shifted to the flexible side in the last three years. In the talk we survey the recent advances in symplectic flexibility in the work of S. Borman, K. Cieliebak, T. Ekholm, E. Murphy, I. Smith, and the author.

References

Peter Albers and Helmut Hofer, On the Weinstein conjecture in higher dimensions, Comment. Math. Helv. 84 (2009), no. 2, 429–436. MR 2495800, DOI 10.4171/CMH/167
Aldo Andreotti and Raghavan Narasimhan, A topological property of Runge pairs, Ann. of Math. (2) 76 (1962), 499–509. MR 140714, DOI 10.2307/1970370
Vladimir Arnol′d, Sur une propriété topologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris 261 (1965), 3719–3722 (French). MR 193645
Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87–161 (French). MR 753131
G.D. Birkhoff, Proof of Poincaré’s geometric theorem, Trans. Amer. Math. Soc., 14(1913), 14–22.
S. Borman, Y. Eliashberg and E. Murphy, Existence and classification of overtwisted contact structures in all dimensions, arXiv:1404.6157.
Frédéric Bourgeois, Odd dimensional tori are contact manifolds, Int. Math. Res. Not. 30 (2002), 1571–1574. MR 1912277, DOI 10.1155/S1073792802205048
Kai Cieliebak and Yakov Eliashberg, From Stein to Weinstein and back, American Mathematical Society Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. MR 3012475, DOI 10.1090/coll/059
K. Cieliebak and Y. Eliashberg, The topology of rationally and polynomially convex domains, Invent. Math., 2014, DOI 10.1007/s00222-014-0511-6.
R. Casals, D. Pancholi and F. Presas, Almost contact 5-folds are contact, 2012, arXiv:1203.2166
Vincent Colin, Emmanuel Giroux, and Ko Honda, On the coarse classification of tight contact structures, Topology and geometry of manifolds (Athens, GA, 2001) Proc. Sympos. Pure Math., vol. 71, Amer. Math. Soc., Providence, RI, 2003, pp. 109–120. MR 2024632, DOI 10.1090/pspum/071/2024632
C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol′d, Invent. Math. 73 (1983), no. 1, 33–49. MR 707347, DOI 10.1007/BF01393824
S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666–705. MR 1438190
Mihai Damian, On the stable Morse number of a closed manifold, Bull. London Math. Soc. 34 (2002), no. 4, 420–430. MR 1897421, DOI 10.1112/S0024609301008955
S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), no. 2, 205–236. MR 1802722
Ferdinand Docquier and Hans Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94–123 (German). MR 148939, DOI 10.1007/BF01360084
T. Ekholm and I. Smith, Exact Lagrangian immersions with a single double point, preprint, arXiv:1111.5932.
Julien Duval and Nessim Sibony, Polynomial convexity, rational convexity, and currents, Duke Math. J. 79 (1995), no. 2, 487–513. MR 1344768, DOI 10.1215/S0012-7094-95-07912-5
Tobias Ekholm and Ivan Smith, Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), no. 1-2, 195–240. MR 3157996, DOI 10.1007/s00208-013-0958-6
Tobias Ekholm, Yakov Eliashberg, Emmy Murphy, and Ivan Smith, Constructing exact Lagrangian immersions with few double points, Geom. Funct. Anal. 23 (2013), no. 6, 1772–1803. MR 3132903, DOI 10.1007/s00039-013-0243-6
Y. Eliashberg, Rigidity of symplectic structures, preprint 1981.
Ya. M. Eliashberg, A theorem on the structure of wave fronts and its application in symplectic topology, Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 65–72, 96 (Russian). MR 911776
Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, DOI 10.1007/BF01393840
Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, DOI 10.1142/S0129167X90000034
Yakov Eliashberg, Filling by holomorphic discs and its applications, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45–67. MR 1171908
Yakov Eliashberg, Contact $3$-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165–192 (English, with French summary). MR 1162559
Y. Eliashberg, Recent progress in symplectic flexibility, The Influence of Solomon Lefschetz in Geometry and Topology: $50$ years of Mathematics in CINVESTAV, (L. Katzarkov, E. Lupercio, and F. Turrubiates, eds.), vol. 621, Contemp. Math., American Mathematical Society, Providence, RI, 2014.
Yakov Eliashberg and Maia Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2009), no. 2, 77–127. MR 2496415
Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, DOI 10.1007/978-3-0346-0425-3_{4}
Yakov Eliashberg and Mikhael Gromov, Convex symplectic manifolds, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 135–162. MR 1128541, DOI 10.1090/pspum/052.2/1128541
Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245, DOI 10.1090/gsm/048
Yakov Eliashberg and Emmy Murphy, Lagrangian caps, Geom. Funct. Anal. 23 (2013), no. 5, 1483–1514. MR 3102911, DOI 10.1007/s00039-013-0239-2
J. Etnyre, Contact structures on 5-manifolds, 2012, arXiv:1210.5208.
John B. Etnyre and Ko Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003), no. 1, 59–74. MR 2004728, DOI 10.1016/S0001-8708(02)00027-0
Franc Forstnerič, Complements of Runge domains and holomorphic hulls, Michigan Math. J. 41 (1994), no. 2, 297–308. MR 1278436, DOI 10.1307/mmj/1029004997
Hansjörg Geiges, Contact structures on $1$-connected $5$-manifolds, Mathematika 38 (1991), no. 2, 303–311 (1992). MR 1147828, DOI 10.1112/S002557930000663X
Hansjörg Geiges, Applications of contact surgery, Topology 36 (1997), no. 6, 1193–1220. MR 1452848, DOI 10.1016/S0040-9383(97)00004-9
Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
H. Geiges and C. B. Thomas, Contact topology and the structure of $5$-manifolds with $\pi _1=Z_2$, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 1167–1188 (English, with English and French summaries). MR 1656012
Hansjörg Geiges and Charles B. Thomas, Contact structures, equivariant spin bordism, and periodic fundamental groups, Math. Ann. 320 (2001), no. 4, 685–708. MR 1857135, DOI 10.1007/PL00004491
Emmanuel Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 405–414 (French, with French summary). MR 1957051
Emmanuel Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), no. 3, 615–689 (French). MR 1779622, DOI 10.1007/s002220000082
A. B. Givental′, Lagrangian imbeddings of surfaces and the open Whitney umbrella, Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 35–41, 96 (Russian). MR 868559
John W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421–450. MR 112161, DOI 10.2307/1970192
M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734 (Russian). MR 0263103
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734 (Russian). MR 0263103
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
Larry Guth, Symplectic embeddings of polydisks, Invent. Math. 172 (2008), no. 3, 477–489. MR 2393077, DOI 10.1007/s00222-007-0103-9
Dmitry Fuchs and Serge Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997), no. 5, 1025–1053. MR 1445553, DOI 10.1016/S0040-9383(96)00035-3
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465, DOI 10.1090/amsip/046.1
Bruno Harris, Some calculations of homotopy groups of symmetric spaces, Trans. Amer. Math. Soc. 106 (1963), 174–184. MR 143216, DOI 10.1090/S0002-9947-1963-0143216-6
Ko Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000), 309–368. MR 1786111, DOI 10.2140/gt.2000.4.309
Ko Honda, On the classification of tight contact structures. II, J. Differential Geom. 55 (2000), no. 1, 83–143. MR 1849027
Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563. MR 1244912, DOI 10.1007/BF01232679
Nicolaas H. Kuiper, On $C^1$-isometric imbeddings. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 545–556, 683–689. MR 0075640
E. Levi, Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse, Annali di Mat oura ed appl. 17, 61–87 (1910).
P. Lisca and G. Matić, Tight contact structures and Seiberg-Witten invariants, Invent. Math. 129 (1997), no. 3, 509–525. MR 1465333, DOI 10.1007/s002220050171
Robert Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 3, ix, 1–15 (French, with English summary). MR 478180
J. Martinet, Formes de contact sur les variétés de dimension $3$, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971, pp. 142–163 (French). MR 0350771
W. S. Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969), 143–156. MR 250331
Mark McLean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), no. 4, 1877–1944. MR 2497314, DOI 10.2140/gt.2009.13.1877
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI 10.1090/S0002-9947-1965-0182927-5
Emmy Murphy, Klaus Niederkrüger, Olga Plamenevskaya, and András I. Stipsicz, Loose Legendrians and the plastikstufe, Geom. Topol. 17 (2013), no. 3, 1791–1814. MR 3073936, DOI 10.2140/gt.2013.17.1791
E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, arXiv:1201.2245.
John Nash, $C^1$ isometric imbeddings, Ann. of Math. (2) 60 (1954), 383–396. MR 65993, DOI 10.2307/1969840
S. Yu. Nemirovskiĭ, Finite unions of balls in $\Bbb C^n$ are rationally convex, Uspekhi Mat. Nauk 63 (2008), no. 2(380), 157–158 (Russian); English transl., Russian Math. Surveys 63 (2008), no. 2, 381–382. MR 2640558, DOI 10.1070/RM2008v063n02ABEH004518
S. Nemirovski and K. Siegel, Rationally convex domains and singular Lagrangian surfaces, in preparation.
Klaus Niederkrüger, The plastikstufe—a generalization of the overtwisted disk to higher dimensions, Algebr. Geom. Topol. 6 (2006), 2473–2508. MR 2286033, DOI 10.2140/agt.2006.6.2473
K. Niederkruger and O. van Koert, Every Contact Manifolds can be given a Nonfillable Contact Structure, Int. Math. Res. Notices, 2009, 4463–4479.
Kiyoshi Oka, Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur, Jpn. J. Math. 23 (1953), 97–155 (1954) (French). MR 71089, DOI 10.4099/jjm1924.23.0_{9}7
H. Poincaré, Sur une théorème de géométrie, Rend. Circ. Mat. Palermo 33 (1912), 375–507.
Lee Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995), no. 1, 155–163. MR 1309974, DOI 10.1007/BF01245177
S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR 153022, DOI 10.2307/2372978
Stephen Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. (2) 69 (1959), 327–344. MR 105117, DOI 10.2307/1970186
Denis Sauvaget, Curiosités lagrangiennes en dimension 4, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 6, 1997–2020 (2005) (French, with English and French summaries). MR 2134231
Edgar Lee Stout, Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2305474, DOI 10.1007/978-0-8176-4538-0
Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117–2202. MR 2350473, DOI 10.2140/gt.2007.11.2117
Leonid Polterovich, Monotone Lagrange submanifolds of linear spaces and the Maslov class in cotangent bundles, Math. Z. 207 (1991), no. 2, 217–222. MR 1109663, DOI 10.1007/BF02571385
L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198–210. MR 1097259, DOI 10.1007/BF01896378
Paul Seidel and Ivan Smith, The symplectic topology of Ramanujam’s surface, Comment. Math. Helv. 80 (2005), no. 4, 859–881. MR 2182703, DOI 10.4171/CMH/37
Clifford Henry Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809–822. MR 1306023, DOI 10.4310/MRL.1994.v1.n6.a15
Ilya Ustilovsky, Infinitely many contact structures on $S^{4m+1}$, Internat. Math. Res. Notices 14 (1999), 781–791. MR 1704176, DOI 10.1155/S1073792899000392
Claude Viterbo, A proof of Weinstein’s conjecture in $\textbf {R}^{2n}$, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 4, 337–356 (English, with French summary). MR 917741
C. T. C. Wall, Geometrical connectivity. I, J. London Math. Soc. (2) 3 (1971), 597–604. MR 290387, DOI 10.1112/jlms/s2-3.4.597
Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, DOI 10.14492/hokmj/1381413841
Hassler Whitney, On the topology of differentiable manifolds, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 101–141. MR 0005300