Recent advances in symplectic flexibility

Eliashberg, Yakov

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

Recent advances in symplectic flexibility

Author: Yakov Eliashberg
Journal: Bull. Amer. Math. Soc. 52 (2015), 1-26
MSC (2010): Primary 53D10, 53D05, 53D12
DOI: https://doi.org/10.1090/S0273-0979-2014-01470-3
Published electronically: August 25, 2014
MathSciNet review: 3286479
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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.

[1] Peter Albers and Helmut Hofer, On the Weinstein conjecture in higher dimensions, Comment. Math. Helv. 84 (2009), no. 2, 429–436. MR 2495800, https://doi.org/10.4171/CMH/167
[2] Aldo Andreotti and Raghavan Narasimhan, A topological property of Runge pairs, Ann. of Math. (2) 76 (1962), 499–509. MR 0140714, https://doi.org/10.2307/1970370
[3] 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 0193645
[4] 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
[5] G.D. Birkhoff, Proof of Poincaré's geometric theorem, Trans. Amer. Math. Soc., 14(1913), 14-22.
[6] S. Borman, Y. Eliashberg and E. Murphy, Existence and classification of overtwisted contact structures in all dimensions, arXiv:1404.6157.
[7] Frédéric Bourgeois, Odd dimensional tori are contact manifolds, Int. Math. Res. Not. 30 (2002), 1571–1574. MR 1912277, https://doi.org/10.1155/S1073792802205048
[8] 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
[9] K. Cieliebak and Y. Eliashberg, The topology of rationally and polynomially convex domains, Invent. Math., 2014, DOI 10.1007/s00222-014-0511-6.
[10] R. Casals, D. Pancholi and F. Presas, Almost contact 5-folds are contact, 2012, arXiv:1203.2166
[11] 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, https://doi.org/10.1090/pspum/071/2024632
[12] 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, https://doi.org/10.1007/BF01393824
[13] S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666–705. MR 1438190
[14] Mihai Damian, On the stable Morse number of a closed manifold, Bull. London Math. Soc. 34 (2002), no. 4, 420–430. MR 1897421, https://doi.org/10.1112/S0024609301008955
[15] S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), no. 2, 205–236. MR 1802722
[16] Ferdinand Docquier and Hans Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94–123 (German). MR 0148939, https://doi.org/10.1007/BF01360084
[17] T. Ekholm and I. Smith, Exact Lagrangian immersions with a single double point, preprint, arXiv:1111.5932.
[18] Julien Duval and Nessim Sibony, Polynomial convexity, rational convexity, and currents, Duke Math. J. 79 (1995), no. 2, 487–513. MR 1344768, https://doi.org/10.1215/S0012-7094-95-07912-5
[19] Tobias Ekholm and Ivan Smith, Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), no. 1-2, 195–240. MR 3157996, https://doi.org/10.1007/s00208-013-0958-6
[20] 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, https://doi.org/10.1007/s00039-013-0243-6
[21] Y. Eliashberg, Rigidity of symplectic structures, preprint 1981.
[22] 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
[23] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, https://doi.org/10.1007/BF01393840
[24] Yakov Eliashberg, Topological characterization of Stein manifolds of dimension >2, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, https://doi.org/10.1142/S0129167X90000034
[25] 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
[26] 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
[27] Y. Eliashberg, Recent progress in symplectic flexibility, The Influence of Solomon Lefschetz in Geometry and Topology: years of Mathematics in CINVESTAV, (L. Katzarkov, E. Lupercio, and F. Turrubiates, eds.), vol. 621, Contemp. Math., American Mathematical Society, Providence, RI, 2014.
[28] Yakov Eliashberg and Maia Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2009), no. 2, 77–127. MR 2496415
[29] 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, https://doi.org/10.1007/978-3-0346-0425-3_4
[30] 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, https://doi.org/10.1090/pspum/052.2/1128541
[31] Y. Eliashberg and N. Mishachev, Introduction to the ℎ-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245
[32] Yakov Eliashberg and Emmy Murphy, Lagrangian caps, Geom. Funct. Anal. 23 (2013), no. 5, 1483–1514. MR 3102911, https://doi.org/10.1007/s00039-013-0239-2
[33] J. Etnyre, Contact structures on 5-manifolds, 2012, arXiv:1210.5208.
[34] John B. Etnyre and Ko Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003), no. 1, 59–74. MR 2004728, https://doi.org/10.1016/S0001-8708(02)00027-0
[35] Franc Forstnerič, Complements of Runge domains and holomorphic hulls, Michigan Math. J. 41 (1994), no. 2, 297–308. MR 1278436, https://doi.org/10.1307/mmj/1029004997
[36] Hansjörg Geiges, Contact structures on 1-connected 5-manifolds, Mathematika 38 (1991), no. 2, 303–311 (1992). MR 1147828, https://doi.org/10.1112/S002557930000663X
[37] Hansjörg Geiges, Applications of contact surgery, Topology 36 (1997), no. 6, 1193–1220. MR 1452848, https://doi.org/10.1016/S0040-9383(97)00004-9
[38] Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738
[39] H. Geiges and C. B. Thomas, Contact topology and the structure of 5-manifolds with 𝜋₁=𝑍₂, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 1167–1188 (English, with English and French summaries). MR 1656012
[40] 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, https://doi.org/10.1007/PL00004491
[41] 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
[42] 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, https://doi.org/10.1007/s002220000082
[43] 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
[44] John W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421–450. MR 0112161, https://doi.org/10.2307/1970192
[45] M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734 (Russian). MR 0263103
[46] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, https://doi.org/10.1007/BF01388806
[47] M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734 (Russian). MR 0263103
[48] 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
[49] Larry Guth, Symplectic embeddings of polydisks, Invent. Math. 172 (2008), no. 3, 477–489. MR 2393077, https://doi.org/10.1007/s00222-007-0103-9
[50] 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, https://doi.org/10.1016/S0040-9383(96)00035-3
[51] 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
[52] Bruno Harris, Some calculations of homotopy groups of symmetric spaces, Trans. Amer. Math. Soc. 106 (1963), 174–184. MR 0143216, https://doi.org/10.1090/S0002-9947-1963-0143216-6
[53] Ko Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000), 309–368. MR 1786111, https://doi.org/10.2140/gt.2000.4.309
[54] Ko Honda, On the classification of tight contact structures. II, J. Differential Geom. 55 (2000), no. 1, 83–143. MR 1849027
[55] Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 0119214, https://doi.org/10.1090/S0002-9947-1959-0119214-4
[56] 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, https://doi.org/10.1007/BF01232679
[57] Nicolaas H. Kuiper, On 𝐶¹-isometric imbeddings. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 545–556, 683–689. MR 0075640
[58] 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).
[59] P. Lisca and G. Matić, Tight contact structures and Seiberg-Witten invariants, Invent. Math. 129 (1997), no. 3, 509–525. MR 1465333, https://doi.org/10.1007/s002220050171
[60] 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 0478180
[61] J. Martinet, Formes de contact sur les variétés de dimension 3, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Springer, Berlin, 1971, pp. 142–163. Lecture Notes in Math., Vol. 209 (French). MR 0350771
[62] W. S. Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969), 143–156. MR 0250331
[63] Mark McLean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), no. 4, 1877–1944. MR 2497314, https://doi.org/10.2140/gt.2009.13.1877
[64] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 0196736, https://doi.org/10.1090/S0002-9904-1966-11484-2
[65] Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 0182927, https://doi.org/10.1090/S0002-9947-1965-0182927-5
[66] 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, https://doi.org/10.2140/gt.2013.17.1791
[67] E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, arXiv:1201.2245.
[68] John Nash, 𝐶¹ isometric imbeddings, Ann. of Math. (2) 60 (1954), 383–396. MR 0065993, https://doi.org/10.2307/1969840
[69] S. Yu. Nemirovskiĭ, Finite unions of balls in ℂⁿ 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, https://doi.org/10.1070/RM2008v063n02ABEH004518
[70] S. Nemirovski and K. Siegel, Rationally convex domains and singular Lagrangian surfaces, in preparation.
[71] Klaus Niederkrüger, The plastikstufe—a generalization of the overtwisted disk to higher dimensions, Algebr. Geom. Topol. 6 (2006), 2473–2508. MR 2286033, https://doi.org/10.2140/agt.2006.6.2473
[72] K. Niederkruger and O. van Koert, Every Contact Manifolds can be given a Nonfillable Contact Structure, Int. Math. Res. Notices, 2009, 4463-4479.
[73] Kiyoshi Oka, Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur, Jap. J. Math. 23 (1953), 97–155 (1954) (French). MR 0071089
[74] H. Poincaré, Sur une théorème de géométrie, Rend. Circ. Mat. Palermo 33 (1912), 375-507.
[75] Lee Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995), no. 1, 155–163. MR 1309974, https://doi.org/10.1007/BF01245177
[76] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR 0153022, https://doi.org/10.2307/2372978
[77] Stephen Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. (2) 69 (1959), 327–344. MR 0105117, https://doi.org/10.2307/1970186
[78] 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
[79] Edgar Lee Stout, Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2305474
[80] Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117–2202. MR 2350473, https://doi.org/10.2140/gt.2007.11.2117
[81] 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, https://doi.org/10.1007/BF02571385
[82] L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198–210. MR 1097259, https://doi.org/10.1007/BF01896378
[83] Paul Seidel and Ivan Smith, The symplectic topology of Ramanujam’s surface, Comment. Math. Helv. 80 (2005), no. 4, 859–881. MR 2182703, https://doi.org/10.4171/CMH/37
[84] Clifford Henry Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809–822. MR 1306023, https://doi.org/10.4310/MRL.1994.v1.n6.a15
[85] Ilya Ustilovsky, Infinitely many contact structures on 𝑆^{4𝑚+1}, Internat. Math. Res. Notices 14 (1999), 781–791. MR 1704176, https://doi.org/10.1155/S1073792899000392
[86] Claude Viterbo, A proof of Weinstein’s conjecture in 𝑅²ⁿ, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 4, 337–356 (English, with French summary). MR 917741
[87] C. T. C. Wall, Geometrical connectivity. I, J. London Math. Soc. (2) 3 (1971), 597–604. MR 0290387, https://doi.org/10.1112/jlms/s2-3.4.597
[88] Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, https://doi.org/10.14492/hokmj/1381413841
[89] Hassler Whitney, On the topology of differentiable manifolds, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 101–141. MR 0005300