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Pseudoholomorphic curves on the LCS-fication of contact manifolds

  • Yong-Geun Oh EMAIL logo and Yasha Savelyev
Published/Copyright: June 3, 2023
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Advances in Geometry
From the journal Advances in Geometry Volume 23 Issue 2

Abstract

For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S1 = Mid (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form

∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ

for the map u = (w, f) : Σ˙→Q×S1for a λ-compatible almost complex structure J and a punctured Riemann surface (Σ˙,j).In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H1(Σ˙,Z)and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of (Q, λ) (more generally on arbitrary locally conformal symplectic manifolds).

Keywords: Locally conformal symplectic manifold; lcs-fication of contact manifold; lcs instanton
MSC 2010: 53D05; 53D40

Funding statement: Oh’s work is supported by the IBS project # IBS-R003-D1.

Acknowledgements

We would like to thank the unknown referee for her/ his careful reading of the paper and pointing out many careless typos and incorrect English expressions which we appreciate very much.

  1. Communicated by: K. Ono

References

[1] C. Abbas, Holomorphic open book decompositions. Duke Math. J 158 (2011), 29–82. MR2794368 Zbl 1230.5307710.1215/00127094-1276301Search in Google Scholar

[2] C. Abbas, K. Cieliebak, H. Hofer, The Weinstein conjecture for planar contact structures in dimension three. Comment. Math. Helv 80 (2005), 771–793. MR2182700 Zbl 1098.5306310.4171/cmh/34Search in Google Scholar

[3] P. Albers, B. Bramham, C. Wendl, On nonseparating contact hypersurfaces in symplectic 4-manifolds. Algebr. Geom. Topol 10 (2010), 697–737. MR2606798 Zbl 1207.3201910.2140/agt.2010.10.697Search in Google Scholar

[4] V. Apostolov, G. Dloussky, Locally conformally symplectic structures on compact non-Kähler complex surfaces. Int. Math. Res. Not no. 9 (2016), 2717–2747. MR3519128 Zbl 1404.3203210.1093/imrn/rnv211Search in Google Scholar

[5] A. Banyaga, Some properties of locally conformal symplectic structures. Comment. Math. Helv 77 (2002), 383–398. MR1915047 Zbl 1020.5305010.1007/s00014-002-8345-zSearch in Google Scholar

[6] A. Banyaga, Examples of non dω-exact locally conformal symplectic forms. J. Geom 87 (2007), 1–13. MR2372512 Zbl 1157.5304010.1007/s00022-006-1849-8Search in Google Scholar

[7] G. Bazzoni, M. Marrero, On locally conformal symplectic manifolds of the first kind, Bull. Sci. Math 187 (2018), 1–57. MR3767426 Zbl 1383.2200410.1016/j.bulsci.2017.10.001Search in Google Scholar

[8] F. Bourgeois, A Morse-Bott approach to contact homology PhD thesis, Stanford University, 2002. MR270329210.1090/fic/035/03Search in Google Scholar

[9] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, Compactness results in symplectic field theory. Geom. Topol 7 (2003), 799–888. MR2026549 Zbl 1131.5331210.2140/gt.2003.7.799Search in Google Scholar

[10] F. Bourgeois, K. Mohnke, Coherent orientations in symplectic field theory. Math. Z 248 (2004), 123–146. MR2092725 Zbl 1060.5308010.1007/s00209-004-0656-xSearch in Google Scholar

[11] S.-F. Chiu, Nonsqueezing property of contact balls. Duke Math. J 166 (2017), 605–655. MR3619302 Zbl 1372.5308710.1215/00127094-3715517Search in Google Scholar

[12] C. Conley, E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math 37 (1984), 207–253. MR733717 Zbl 0559.5801910.1002/cpa.3160370204Search in Google Scholar

[13] Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory. In: GAFA 2000 (Tel Aviv, 1999), Part II, 560–673, Birkhäuser 2000. MR1826267 Zbl 0989.8111410.1007/978-3-0346-0425-3_4Search in Google Scholar

[14] Y. Eliashberg, S. S. Kim, L. Polterovich, Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol 10 (2006), 1635–1747. MR2284048 Zbl 1134.5304410.2140/gt.2006.10.1635Search in Google Scholar

[15] Y. Eliashberg, E. Murphy, Making cobordisms symplectic. Preprint 2015, arXiv:1504.06312 [math.SG]Search in Google Scholar

[16] M. Fraser, Contact non-squeezing at large scale in ℝ2n × S1Internat. J. Math 27 (2016), 1650107, 25 pages. MR3589657 Zbl 1357.5310010.1142/S0129167X1650107XSearch in Google Scholar

[17] P. Gauduchon, Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B (7) 11 (1997), 257–288. MR1456265 Zbl 0876.53015Search in Google Scholar

[18] S. Haller, T. Rybicki, On the group of diffeomorphisms preserving a locally conformal symplectic structure. Ann. Global Anal. Geom 17 (1999), 475–502. MR1715157 Zbl 0940.5304410.1023/A:1006650124434Search in Google Scholar

[19] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math 114 (1993), 515–563. MR1244912 Zbl 0797.5802310.1007/BF01232679Search in Google Scholar

[20] H. Hofer, Holomorphic curves and real three-dimensional dynamics. In: GAFA 2000 (Tel Aviv, 1999), Part II, 674–704, Birkhäuser 2000. MR1826268 Zbl 1161.5336210.1007/978-3-0346-0425-3_5Search in Google Scholar

[21] H. Hofer, C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres. Comm. Pure Appl. Math 45 (1992), 583–622. MR1162367 Zbl 0773.5802110.1002/cpa.3160450504Search in Google Scholar

[22] S. Kobayashi, Natural connections in almost complex manifolds. In: Explorations in complex and Riemannian geometry volume 332 of Contemp. Math 153–169, Amer. Math. Soc. 2003. MR2018338 Zbl 1052.5303210.1090/conm/332/05935Search in Google Scholar

[23] H. V. Lê, Y.-G. Oh, Deformations of coisotropic submanifolds in locally conformal symplectic manifolds. Asian J. Math 20 (2016), 553–596. MR3528832 Zbl 1352.5307210.4310/AJM.2016.v20.n3.a7Search in Google Scholar

[24] H.-C. Lee, A kind of even-dimensional differential geometry and its application to exterior calculus. Amer. J. Math 65 (1943), 433–438. MR8495 Zbl 0060.3830210.2307/2371967Search in Google Scholar

[25] R. B. Lockhart, R. C. McOwen, Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 12 (1985), 409–447. MR837256 Zbl 0615.58048Search in Google Scholar

[26] Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings. Internat. Math. Res. Notices no. 7 (1996), 305–346. MR1389956 Zbl 0858.5801710.1155/S1073792896000219Search in Google Scholar

[27] Y.-G. Oh, Symplectic topology and Floer homology. Vol 1, volume 28 of New Mathematical Monographs Cambridge Univ. Press 2015. MR3443239 Zbl 1338.53004Search in Google Scholar

[28] Y.-G. Oh, Analysis of contact Cauchy–Riemann maps III: energy, bubbling and Fredholm theory. Bull. Math. Sci (2022) published online, 22501, 61 pages. DOI 10.1142/S166436072250011410.1142/S1664360722500114Search in Google Scholar

[29] Y.-G. Oh, Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition. Preprint 2021, arXiv:2103.15390 [math.SG]Search in Google Scholar

[30] Y.-G. Oh, R. Wang, Canonical connection on contact manifolds. In: Real and complex submanifolds volume 106 of Springer Proc. Math. Stat 43–63, Springer 2014. MR3333367 Zbl 1323.5308610.1007/978-4-431-55215-4_5Search in Google Scholar

[31] Y.-G. Oh, R. Wang, Analysis of contact Cauchy-Riemann maps I: a priori Ck estimates and asymptotic convergence. Osaka J. Math 55 (2018), 647–679. MR3862779 Zbl 1454.53074Search in Google Scholar

[32] Y.-G. Oh, R. Wang, Analysis of contact Cauchy-Riemann maps II: Canonical neighborhoods and exponential convergence for the Morse-Bott case. Nagoya Math. J 231 (2018), 128–223. MR3845592 Zbl 1410.5308010.1017/nmj.2017.17Search in Google Scholar

[33] J. Robbin, D. Salamon, The Maslov index for paths. Topology 32 (1993), 827–844. MR1241874 Zbl 0798.5801810.1016/0040-9383(93)90052-WSearch in Google Scholar

[34] J. Sacks, K. Uhlenbeck, The existence of minimal immersions of 2-spheres. Ann. of Math (2) 113 (1981), 1–24. MR604040 Zbl 0462.5801410.2307/1971131Search in Google Scholar

[35] I. Vaisman, Locally conformal symplectic manifolds. Internat. J. Math. Math. Sci 8 (1985), 521–536. MR809073 Zbl 0585.5303010.1155/S0161171285000564Search in Google Scholar
Received: 2021-11-10
Revised: 2022-01-21
Published Online: 2023-06-03
Published in Print: 2023-05-25

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Pseudoholomorphic curves on the LCS-fication of contact manifolds
  3. Shellable tilings on relative simplicial complexes and their h-vectors
  4. On projections of free semialgebraic sets
  5. Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi–Yau varieties
  6. Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids
  7. Open books and embeddings of smooth and contact manifolds
  8. Fano fourfolds having a prime divisor of Picard number 1
  9. Characterizations of symplectic polar spaces
  10. Chow groups of Gushel–Mukai fivefolds
https://doi.org/10.1515/advgeom-2023-0004
Keywords for this article
Locally conformal symplectic manifold; lcs-fication of contact manifold; lcs instanton

Articles in the same Issue

  1. Frontmatter
  2. Pseudoholomorphic curves on the LCS-fication of contact manifolds
  3. Shellable tilings on relative simplicial complexes and their h-vectors
  4. On projections of free semialgebraic sets
  5. Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi–Yau varieties
  6. Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids
  7. Open books and embeddings of smooth and contact manifolds
  8. Fano fourfolds having a prime divisor of Picard number 1
  9. Characterizations of symplectic polar spaces
  10. Chow groups of Gushel–Mukai fivefolds
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