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.

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 https://doi.org/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 https://doi.org/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 https://doi.org/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

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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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

Yakov Eliashberg and Emmy Murphy, Lagrangian caps, Geom. Funct. Anal. 23 (2013), no. 5, 1483–1514. MR 3102911, DOI https://doi.org/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 https://doi.org/10.1016/S0001-8708%2802%2900027-0

Franc Forstnerič, Complements of Runge domains and holomorphic hulls, Michigan Math. J. 41 (1994), no. 2, 297–308. MR 1278436, DOI https://doi.org/10.1307/mmj/1029004997

Hansjörg Geiges, Contact structures on $1$-connected $5$-manifolds, Mathematika 38 (1991), no. 2, 303–311 (1992). MR 1147828, DOI https://doi.org/10.1112/S002557930000663X

Hansjörg Geiges, Applications of contact surgery, Topology 36 (1997), no. 6, 1193–1220. MR 1452848, DOI https://doi.org/10.1016/S0040-9383%2897%2900004-9

Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738

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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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

Larry Guth, Symplectic embeddings of polydisks, Invent. Math. 172 (2008), no. 3, 477–489. MR 2393077, DOI https://doi.org/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 https://doi.org/10.1016/S0040-9383%2896%2900035-3

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

Bruno Harris, Some calculations of homotopy groups of symmetric spaces, Trans. Amer. Math. Soc. 106 (1963), 174–184. MR 143216, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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), Springer, Berlin, 1971, pp. 142–163. Lecture Notes in Math., Vol. 209 (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 https://doi.org/10.2140/gt.2009.13.1877

J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.4099/jjm1924.23.0_97

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 https://doi.org/10.1007/BF01245177

S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR 153022, DOI https://doi.org/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 https://doi.org/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

Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117–2202. MR 2350473, DOI https://doi.org/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 https://doi.org/10.1007/BF02571385

L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198–210. MR 1097259, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1155/S1073792899000392

Claude Viterbo, A proof of Weinstein’s conjecture in ${\bf R}^{2n}$, 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 https://doi.org/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 https://doi.org/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