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
DOI:
https://doi.org/10.1090/S0273-0979-09-01282-8
Published electronically:
October 29, 2009
MathSciNet review:
2566446
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- Casim Abbas, The chord problem and a new method of filling by pseudoholomorphic curves, Int. Math. Res. Not. 18 (2004), 913–927. MR 2037757, DOI https://doi.org/10.1155/S1073792804132236
- 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, DOI https://doi.org/10.4171/CMH/34
- 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
- Vladimir I. Arnold and Boris A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. MR 1612569
- D. Auroux, La conjecture de Weinstein en dimension $3$ [d’après C.H. Taubes], Séminaire Bourbaki, 2008-2009, no. 1002.
- 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
- Raoul Bott, Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 99–114 (1989). MR 1001450
- 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
- Frédéric Bourgeois, A Morse-Bott approach to contact homology, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001) Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, pp. 55–77. MR 1969267
- 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, DOI https://doi.org/10.3934/jmd.2007.1.597
- F. Bourgeois, T. Ekholm, and Y. Eliashberg, A Legendrian surgery long exact sequence for linearized contact homology, in preparation.
- F. Bourgeois and K. Niederkrüger, Towards a good definition of algebraically overtwisted, arXiv:0709.3415
- 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
- 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, DOI https://doi.org/10.1007/s10240-009-0022-y
- V. Colin and K. Honda, Reeb vector fields and open book decompositions, arXiv:0809.5088.
- 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, DOI https://doi.org/10.1090/S0273-0979-96-00625-8
- 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, DOI https://doi.org/10.2140/gtm.1999.2.87
- 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, 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
- 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
- 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, DOI https://doi.org/10.1090/pspum/071/2024631
- 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
- 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, DOI https://doi.org/10.1112/blms/26.1.88
- Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738
- 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, DOI https://doi.org/10.1007/0-8176-4419-9_6
- Viktor L. Ginzburg and Başak Z. Gürel, A $C^2$-smooth counterexample to the Hamiltonian Seifert conjecture in $\Bbb R^4$, Ann. of Math. (2) 158 (2003), no. 3, 953–976. MR 2031857, DOI https://doi.org/10.4007/annals.2003.158.953
- 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
- J. Harrison, $C^2$ counterexamples to the Seifert conjecture, Topology 27 (1988), no. 3, 249–278. MR 963630, DOI https://doi.org/10.1016/0040-9383%2888%2990009-2
- 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
- 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, DOI https://doi.org/10.4007/annals.2003.157.125
- 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
- M. Hutchings, The embedded contact homology index revisited, arXiv:0805.1240, to appear in the Yashafest proceedings.
- Michael Hutchings and Michael Sullivan, Rounding corners of polygons and the embedded contact homology of $T^3$, Geom. Topol. 10 (2006), 169–266. MR 2207793, DOI https://doi.org/10.2140/gt.2006.10.169
- 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, DOI https://doi.org/10.1090/pcms/007/04
- 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
- Michael Hutchings and Clifford Henry Taubes, The Weinstein conjecture for stable Hamiltonian structures, Geom. Topol. 13 (2009), no. 2, 901–941. MR 2470966, DOI https://doi.org/10.2140/gt.2009.13.901
- Arthur Jaffe and Clifford Taubes, Vortices and monopoles, Progress in Physics, vol. 2, Birkhäuser, Boston, Mass., 1980. Structure of static gauge theories. MR 614447
- Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043
- Greg Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996), no. 1, 70–97. MR 1371679, DOI https://doi.org/10.1007/BF02566410
- Greg Kuperberg and Krystyna Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. of Math. (2) 143 (1996), no. 3, 547–576. MR 1394969, DOI https://doi.org/10.2307/2118536
- Krystyna Kuperberg, A smooth counterexample to the Seifert conjecture, Ann. of Math. (2) 140 (1994), no. 3, 723–732. MR 1307902, DOI https://doi.org/10.2307/2118623
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
- Y-J. Lee and C.H. Taubes, Periodic Floer homology and Seiberg-Witten Floer cohomology, arXiv:0906.0383.
- MSRI Hot Topics Workshop, Contact structures, dynamics and the Seiberg-Witten equations in dimension $3$, June 2008, videos at www.msri.org.
- 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
- Guowu Meng and Clifford Henry Taubes, $\underline {\rm SW}=$ Milnor torsion, Math. Res. Lett. 3 (1996), no. 5, 661–674. MR 1418579, DOI https://doi.org/10.4310/MRL.1996.v3.n5.a8
- Klaus Mohnke, Holomorphic disks and the chord conjecture, Ann. of Math. (2) 154 (2001), no. 1, 219–222. MR 1847594, DOI https://doi.org/10.2307/3062116
- 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
- Y-G. Oh and K. Zhu, Embedding property of $J$-holomorphic curves in Calabi-Yau manifolds for generic $J$, arXiv:0805.3581.
- 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, DOI https://doi.org/10.4007/annals.2004.159.1027
- Peter Ozsváth and Zoltán Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326–400. MR 2222356, DOI https://doi.org/10.1016/j.aim.2005.03.014
- Tim Perutz, Lagrangian matching invariants for fibred four-manifolds. I, Geom. Topol. 11 (2007), 759–828. MR 2302502, DOI https://doi.org/10.2140/gt.2007.11.759
- Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 467823, DOI https://doi.org/10.1002/cpa.3160310203
- A. Rechtman, Existence of periodic orbits for geodesible vector fields on closed $3$-manifolds, arXiv:0904.2719.
- Joel Robbin and Dietmar Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33. MR 1331677, DOI https://doi.org/10.1112/blms/27.1.1
- Matthias Schwarz, Morse homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. MR 1239174
- Matthias Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), no. 2, 419–461. MR 1755825, DOI https://doi.org/10.2140/pjm.2000.193.419
- Paul A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Ann. of Math. (2) 100 (1974), 386–400. MR 356086, DOI https://doi.org/10.2307/1971077
- Clifford Henry Taubes, ${\rm Gr}\Longrightarrow {\rm SW}$: from pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differential Geom. 51 (1999), no. 2, 203–334. MR 1728301
- 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.
- 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
- 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
- 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, DOI https://doi.org/10.2140/gt.2009.13.1337
- Clifford Henry Taubes, Asymptotic spectral flow for Dirac operators, Comm. Anal. Geom. 15 (2007), no. 3, 569–587. MR 2379805
- C.H. Taubes, An observation concerning uniquely ergodic vector fields on $3$-manifolds, arXiv:0811.3983.
- C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology I, arXiv:0811.3985.
- C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology II-IV, preprints, 2008.
- C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology V, preprint, 2008.
- Vladimir Turaev, A combinatorial formulation for the Seiberg-Witten invariants of $3$-manifolds, Math. Res. Lett. 5 (1998), no. 5, 583–598. MR 1666856, DOI https://doi.org/10.4310/MRL.1998.v5.n5.a3
- Michael Usher, Vortices and a TQFT for Lefschetz fibrations on 4-manifolds, Algebr. Geom. Topol. 6 (2006), 1677–1743. MR 2263047, DOI https://doi.org/10.2140/agt.2006.6.1677
- 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
- Alan Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), no. 3, 507–518. MR 512430, DOI https://doi.org/10.2307/1971185
- Alan Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979), no. 3, 353–358. MR 543704, DOI https://doi.org/10.1016/0022-0396%2879%2990070-6
- Edward Witten, From superconductors and four-manifolds to weak interactions, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 361–391. MR 2318156, DOI https://doi.org/10.1090/S0273-0979-07-01167-6
- 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
Retrieve articles in Bulletin of the American Mathematical Society with MSC (2010): 57R17, 57R57, 53D40
Retrieve articles in all journals with MSC (2010): 57R17, 57R57, 53D40
Additional Information
Michael Hutchings
Affiliation:
Mathematics Department, 970 Evans Hall, University of California, Berkeley, California 94720
Email:
hutching@math.berkeley.edu
Received by editor(s):
June 11, 2009
Received by editor(s) in revised form:
August 26, 2009
Published electronically:
October 29, 2009
Additional Notes:
Partially supported by NSF grant DMS-0806037
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.