×

The Seiberg-Witten equations and the Weinstein conjecture. (English) Zbl 1135.57015

Let \(\mathcal{M}\) be a compact, orientable 3-dimensional manifold and let \(\alpha\) be a contact form. The kernel of the 1-form \(\alpha\) yields an oriented 2-plane subbundle \(K^{-1}\) of \(TM\). Weinstein conjectured that the associated Reeb vector field has at least one closed integral curve. The author proves this conjecture. Namely, for a fixed 2-dimensional integral cohomology class \(\mathcal{B}\) that differs from half the first Chern class of \(K\) by a torsion element, there are a nonempty set of closed integral curves of the Reeb vector field and a positive integer weight assigned to each curve in this set such that the resulting formal weighted sum of loops represents the Poincaré dual of \(\mathcal{B}\). To prove the theorem he uses ideas identifying the Seiberg-Witten and Gromov invariants of a compact, symplectic 4-manifold, and estimating the spectral flow of a family of Dirac operators on the 3-manifold \(\mathcal{M}\) as follows : Deform the Seiberg-Witten equations on the 3-manifold by adding a constant multiple \(r\) of \(-id\alpha\) to the curvature equation. Consider a sequence of values of \(r\) that tend to \(\infty\) and a corresponding sequence of solutions to the resulting equations. Under optimal circumstances, the spinor component of a solution to a large \(r\) of the equations is nearly zero on a set that closely approximates a closed integral curve of the Reeb vector field. Here the Seiberg-Witten Floer homology described by P. Kronheimer and T. Mrowka is used to prove that all larger \(r\) versions of the equations have solutions. As \(r\) goes to \(\infty\) along the sequence, the author shows that there is a subsequence of such sets that limits to the desired closed integral curve.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] C Abbas, K Cieliebak, H Hofer, The Weinstein conjecture for planar contact structures in dimension three, Comment. Math. Helv. 80 (2005) 771 · Zbl 1098.53063 · doi:10.4171/CMH/34
[2] M Berger, P Gauduchon, E Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics 194, Springer (1971) · Zbl 0223.53034
[3] N Berline, E Getzler, M Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer (2004) · Zbl 1037.58015
[4] W Chen, Pseudo-holomorphic curves and the Weinstein conjecture, Comm. Anal. Geom. 8 (2000) 115 · Zbl 0978.53135
[5] S Y Cheng, P Li, Heat kernel estimates and lower bound of eigenvalues, Comment. Math. Helv. 56 (1981) 327 · Zbl 0484.53034 · doi:10.1007/BF02566216
[6] V Colin, K Honda, Reeb vector fields and open book decompositions I: the periodic case, preprint (2005) · Zbl 1266.57013 · doi:10.4171/JEMS/365
[7] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560 · Zbl 0989.81114
[8] D T Gay, Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol. 10 (2006) 1749 · Zbl 1129.53061 · doi:10.2140/gt.2006.10.1749
[9] H Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 515 · Zbl 0797.58023 · doi:10.1007/BF01232679
[10] H Hofer, Dynamics, topology, and holomorphic curves (1998) 255 · Zbl 0908.58020
[11] H Hofer, Holomorphic curves and dynamics in dimension three, IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 35 · Zbl 1004.53062
[12] H Hofer, Holomorphic curves and real three-dimensional dynamics, Geom. Funct. Anal. (2000) 674 · Zbl 1161.53362
[13] K Honda, The topology and geometry of contact structures in dimension three, Eur. Math. Soc., Zürich (2006) 705 · Zbl 1099.57011
[14] M Hutchings, M Sullivan, Rounding corners of polygons and the embedded contact homology of \(T^3\), Geom. Topol. 10 (2006) 169 · Zbl 1101.53053 · doi:10.2140/gt.2006.10.169
[15] A Jaffe, C Taubes, Vortices and monopoles, Progress in Physics 2, Birkhäuser (1980) · Zbl 0457.53034
[16] T Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften 132, Springer New York, New York (1966) · Zbl 0148.12601
[17] P. Kronheimer, T. Mrowka, Monopoles and Three-Manifolds, New Mathematical Monographs 10, Cambridge University Press (2007) 770 · Zbl 1158.57002
[18] S. Molchanov, Diffusion process in Riemannian geometry, Uspekhi Mat. Nauk 30 (1975) 1 · Zbl 0315.53026 · doi:10.1070/RM1975v030n01ABEH001400
[19] C B Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften 130, Springer New York, New York (1966) · Zbl 0142.38701
[20] Y G Oh, Floer mini-max theory, the Cerf diagram, and the spectral invariants · Zbl 1180.53084 · doi:10.4134/JKMS.2009.46.2.363
[21] T H Parker, Geodesics and approximate heat kernels
[22] M Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000) 419 · Zbl 1023.57020 · doi:10.2140/pjm.2000.193.419
[23] R T Seeley, Complex powers of an elliptic operator, Amer. Math. Soc. (1967) 288 · Zbl 0159.15504
[24] S Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965) 861 · Zbl 0143.35301 · doi:10.2307/2373250
[25] C H Taubes, Asymptotic spectral flow for Dirac operators, to appear in Commun. Analysis and Geometry
[26] C H Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Lett. 2 (1995) 221 · Zbl 0854.57020 · doi:10.4310/MRL.1995.v2.n2.a10
[27] C H Taubes, \(\mathrm{Gr}{\Rightarrow}\mathrm{SW}\): from pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differential Geom. 51 (1999) 203 · Zbl 1036.53066
[28] C H Taubes, \(\mathrm{SW}{\Rightarrow}\mathrm{Gr}\): from the Seiberg-Witten equations to pseudo-holomorphic curves (editor C H Taubes), First Int. Press Lect. Ser. 2, Int. Press, Somerville, MA (2000) 1
[29] C H Taubes, Seiberg Witten and Gromov invariants for symplectic 4-manifolds, First International Press Lecture Series 2, International Press (2000) · Zbl 0967.57001
[30] A Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979) 353 · Zbl 0388.58020 · doi:10.1016/0022-0396(79)90070-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.