Taubes’s proof of the Weinstein conjecture in dimension three

Hutchings, Michael

doi:10.1090/S0273-0979-09-01282-8

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Taubes's proof of the Weinstein conjecture in dimension three

Author: Michael Hutchings
Journal: Bull. Amer. Math. Soc. 47 (2010), 73-125
MSC (2010): Primary 57R17, 57R57, 53D40
Posted: October 29, 2009
MathSciNet review: 2566446
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Abstract: Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? The answer is no, according to counterexamples by K. Kuperberg and others. On the other hand, there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes's proof, and the bigger picture into which it fits.

1. Casim Abbas, The chord problem and a new method of filling by pseudoholomorphic curves, Int. Math. Res. Not. 18 (2004), 913–927. MR 2037757 (2005a:53150), http://dx.doi.org/10.1155/S1073792804132236
2. Casim Abbas, Kai Cieliebak, and Helmut Hofer, The Weinstein conjecture for planar contact structures in dimension three, Comment. Math. Helv. 80 (2005), no. 4, 771–793. MR 2182700 (2006g:53135), http://dx.doi.org/10.4171/CMH/34
3. Peter Albers and Helmut Hofer, On the Weinstein conjecture in higher dimensions, Comment. Math. Helv. 84 (2009), no. 2, 429–436. MR 2495800 (2010b:53152), http://dx.doi.org/10.4171/CMH/167
4. Vladimir I. Arnold and Boris A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. MR 1612569 (99b:58002)
5. D. Auroux, La conjecture de Weinstein en dimension [d'après C.H. Taubes], Séminaire Bourbaki, 2008-2009, no. 1002.
6. Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720 (94e:58130)
7. Raoul Bott, Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 99–114 (1989). MR 1001450 (90f:58027)
8. Frédéric Bourgeois, Odd dimensional tori are contact manifolds, Int. Math. Res. Not. 30 (2002), 1571–1574. MR 1912277 (2003f:53157), http://dx.doi.org/10.1155/S1073792802205048
9. Frédéric Bourgeois, A Morse-Bott approach to contact homology, (Toronto, ON/Montreal, QC, 2001) Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, pp. 55–77. MR 1969267 (2004a:53109)
10. Frédéric Bourgeois, Kai Cieliebak, and Tobias Ekholm, A note on Reeb dynamics on the tight 3-sphere, J. Mod. Dyn. 1 (2007), no. 4, 597–613. MR 2342700 (2008h:53150), http://dx.doi.org/10.3934/jmd.2007.1.597
11. F. Bourgeois, T. Ekholm, and Y. Eliashberg, A Legendrian surgery long exact sequence for linearized contact homology, in preparation.
12. F. Bourgeois and K. Niederkrüger, Towards a good definition of algebraically overtwisted, arXiv:0709.3415
13. K. Cieliebak and K. Mohnke, Compactness for punctured holomorphic curves, J. Symplectic Geom. 3 (2005), no. 4, 589–654. Conference on Symplectic Topology. MR 2235856 (2007b:53181)
14. Vincent Colin, Emmanuel Giroux, and Ko Honda, Finitude homotopique et isotopique des structures de contact tendues, Publ. Math. Inst. Hautes Études Sci. 109 (2009), 245–293 (French, with French summary). MR 2511589 (2010g:53167), http://dx.doi.org/10.1007/s10240-009-0022-y
15. V. Colin and K. Honda, Reeb vector fields and open book decompositions, arXiv:0809.5088.
16. S. K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 45–70. MR 1339810 (96k:57033), http://dx.doi.org/10.1090/S0273-0979-96-00625-8
17. S. K. Donaldson, Topological field theories and formulae of Casson and Meng-Taubes, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 87–102. MR 1734402 (2001f:57033), http://dx.doi.org/10.2140/gtm.1999.2.87
18. Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310 (90k:53064), http://dx.doi.org/10.1007/BF01393840
19. 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 (93g:53060)
20. 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 (2002e:53136), http://dx.doi.org/10.1007/978-3-0346-0425-3_4
21. John B. Etnyre, Introductory lectures on contact geometry, Topology and geometry of manifolds (Athens, GA, 2001) Proc. Sympos. Pure Math., vol. 71, Amer. Math. Soc., Providence, RI, 2003, pp. 81–107. MR 2024631 (2005b:53139)
22. John B. Etnyre, Lectures on open book decompositions and contact structures, Floer homology, gauge theory, and low-dimensional topology, Clay Math. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 2006, pp. 103–141. MR 2249250 (2007g:57042)
23. Oscar García-Prada, A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc. 26 (1994), no. 1, 88–96. MR 1246476 (95d:53025), http://dx.doi.org/10.1112/blms/26.1.88
24. Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738 (2008m:57064)
25. Viktor L. Ginzburg, The Weinstein conjecture and theorems of nearby and almost existence, The breadth of symplectic and Poisson geometry, Progr. Math., vol. 232, Birkhäuser Boston, Boston, MA, 2005, pp. 139–172. MR 2103006 (2005m:53168), http://dx.doi.org/10.1007/0-8176-4419-9_6
26. Viktor L. Ginzburg and Başak Z. Gürel, A 𝐶²-smooth counterexample to the Hamiltonian Seifert conjecture in ℝ⁴, Ann. of Math. (2) 158 (2003), no. 3, 953–976. MR 2031857 (2005e:37133), http://dx.doi.org/10.4007/annals.2003.158.953
27. Emmanuel Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 405–414 (French, with French summary). MR 1957051 (2004c:53144)
28. J. Harrison, 𝐶² counterexamples to the Seifert conjecture, Topology 27 (1988), no. 3, 249–278. MR 963630 (89m:58167), http://dx.doi.org/10.1016/0040-9383(88)90009-2
29. 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 (94j:58064), http://dx.doi.org/10.1007/BF01232679
30. H. Hofer, K. Wysocki, and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003), no. 1, 125–255. MR 1954266 (2004a:53108), http://dx.doi.org/10.4007/annals.2003.157.125
31. Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1306732 (96g:58001)
32. M. Hutchings, The embedded contact homology index revisited, arXiv:0805.1240, to appear in the Yashafest proceedings.
33. Michael Hutchings and Michael Sullivan, Rounding corners of polygons and the embedded contact homology of 𝑇³, Geom. Topol. 10 (2006), 169–266. MR 2207793 (2006k:53150), http://dx.doi.org/10.2140/gt.2006.10.169
34. Michael Hutchings and Clifford Henry Taubes, An introduction to the Seiberg-Witten equations on symplectic manifolds, Symplectic geometry and topology (Park City, UT, 1997) IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, 1999, pp. 103–142. MR 1702943 (2000j:57066)
35. Michael Hutchings and Clifford Henry Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. I, J. Symplectic Geom. 5 (2007), no. 1, 43–137. MR 2371184 (2008k:53201)
36. Michael Hutchings and Clifford Henry Taubes, The Weinstein conjecture for stable Hamiltonian structures, Geom. Topol. 13 (2009), no. 2, 901–941. MR 2470966 (2010h:53139), http://dx.doi.org/10.2140/gt.2009.13.901
37. Arthur Jaffe and Clifford Taubes, Vortices and monopoles, Progress in Physics, vol. 2, Birkhäuser Boston, Mass., 1980. Structure of static gauge theories. MR 614447 (82m:81051)
38. Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043 (2009f:57049)
39. Greg Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996), no. 1, 70–97. MR 1371679 (96m:58199), http://dx.doi.org/10.1007/BF02566410
40. Greg Kuperberg and Krystyna Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. of Math. (2) 143 (1996), no. 3, 547–576. MR 1394969 (97k:57031a), http://dx.doi.org/10.2307/2118536
41. Krystyna Kuperberg, A smooth counterexample to the Seifert conjecture, Ann. of Math. (2) 140 (1994), no. 3, 723–732. MR 1307902 (95g:57040), http://dx.doi.org/10.2307/2118623
42. H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)
43. Y-J. Lee and C.H. Taubes, Periodic Floer homology and Seiberg-Witten Floer cohomology, arXiv:0906.0383.
44. MSRI Hot Topics Workshop, Contact structures, dynamics and the Seiberg-Witten equations in dimension , June 2008, videos at www.msri.org.
45. Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998. MR 1698616 (2000g:53098)
46. Guowu Meng and Clifford Henry Taubes, \𝑢𝑛𝑑𝑒𝑟𝑙𝑖𝑛𝑒{𝑆𝑊}= Milnor torsion, Math. Res. Lett. 3 (1996), no. 5, 661–674. MR 1418579 (98j:57049)
47. Klaus Mohnke, Holomorphic disks and the chord conjecture, Ann. of Math. (2) 154 (2001), no. 1, 219–222. MR 1847594 (2002f:53147), http://dx.doi.org/10.2307/3062116
48. John W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, vol. 44, Princeton University Press, Princeton, NJ, 1996. MR 1367507 (97d:57042)
49. Y-G. Oh and K. Zhu, Embedding property of -holomorphic curves in Calabi-Yau manifolds for generic , arXiv:0805.3581.
50. Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019 (2006b:57016), http://dx.doi.org/10.4007/annals.2004.159.1027
51. Peter Ozsváth and Zoltán Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326–400. MR 2222356 (2007i:57029), http://dx.doi.org/10.1016/j.aim.2005.03.014
52. Tim Perutz, Lagrangian matching invariants for fibred four-manifolds. I, Geom. Topol. 11 (2007), 759–828. MR 2302502 (2008e:53175), http://dx.doi.org/10.2140/gt.2007.11.759
53. Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 0467823 (57 #7674)
54. A. Rechtman, Existence of periodic orbits for geodesible vector fields on closed -manifolds, arXiv:0904.2719.
55. Joel Robbin and Dietmar Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33. MR 1331677 (96d:58021), http://dx.doi.org/10.1112/blms/27.1.1
56. Matthias Schwarz, Morse homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. MR 1239174 (95a:58022)
57. Matthias Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), no. 2, 419–461. MR 1755825 (2001c:53113), http://dx.doi.org/10.2140/pjm.2000.193.419
58. Paul A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math. (2) 100 (1974), 386–400. MR 0356086 (50 #8557)
59. Clifford Henry Taubes, 𝐺𝑟⟹𝑆𝑊: from pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differential Geom. 51 (1999), no. 2, 203–334. MR 1728301 (2000i:53123)
60. C.H. Taubes, $SW\Rightarrow Gr$ : From the Seiberg-Witten equations to pseudo-holomorphic curves, in ``Seiberg-Witten and Gromov invariants for symplectic 4-manifolds'', Internat. Press, 2000.
61. Clifford Henry Taubes, Seiberg Witten and Gromov invariants for symplectic 4-manifolds, First International Press Lecture Series, vol. 2, International Press, Somerville, MA, 2000. Edited by Richard Wentworth. MR 1798809 (2002j:53115)
62. Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117–2202. MR 2350473 (2009b:57055), http://dx.doi.org/10.2140/gt.2007.11.2117
63. Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture. II. More closed integral curves of the Reeb vector field, Geom. Topol. 13 (2009), no. 3, 1337–1417. MR 2496048 (2010f:57027), http://dx.doi.org/10.2140/gt.2009.13.1337
64. Clifford Henry Taubes, Asymptotic spectral flow for Dirac operators, Comm. Anal. Geom. 15 (2007), no. 3, 569–587. MR 2379805 (2009a:58025)
65. C.H. Taubes, An observation concerning uniquely ergodic vector fields on -manifolds, arXiv:0811.3983.
66. C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology I, arXiv:0811.3985.
67. C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology II-IV, preprints, 2008.
68. C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology V, preprint, 2008.
69. Vladimir Turaev, A combinatorial formulation for the Seiberg-Witten invariants of 3-manifolds, Math. Res. Lett. 5 (1998), no. 5, 583–598. MR 1666856 (2000c:57027)
70. Michael Usher, Vortices and a TQFT for Lefschetz fibrations on 4-manifolds, Algebr. Geom. Topol. 6 (2006), 1677–1743 (electronic). MR 2263047 (2007g:53101), http://dx.doi.org/10.2140/agt.2006.6.1677
71. 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 (89d:58048)
72. Alan Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), no. 3, 507–518. MR 512430 (80g:58034), http://dx.doi.org/10.2307/1971185
73. Alan Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979), no. 3, 353–358. MR 543704 (81a:58030b), http://dx.doi.org/10.1016/0022-0396(79)90070-6
74. Edward Witten, From superconductors and four-manifolds to weak interactions, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 361–391 (electronic). MR 2318156 (2008e:58021), http://dx.doi.org/10.1090/S0273-0979-07-01167-6
75. E. Zehnder, Remarks on periodic solutions on hypersurfaces, Periodic solutions of Hamiltonian systems and related topics (Il Ciocco, 1986), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 209, Reidel, Dordrecht, 1987, pp. 267–279. MR 920629 (89b:58083)

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