Skip to main content
SpringerLink
Log in

Positive factorizations of pseudoperiodic homeomorphisms

Mathematische Annalen volume 379, pages 1173–1203 (2021)Cite this article

Abstract

We generalize a classical result concerning smooth germs of surfaces, by proving that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs admit a positive factorization. As a consequence of this and a topological characterization of these monodromies by Anne Pichon, we conclude that a pseudoperiodic homeomorphism on an oriented surface with boundary with positive fractional Dehn twist coefficients and screw numbers, admits a positive factorization. We use the main theorem to give a sufficiency criterion for certain pseudoperiodic homeomorphisms with negative screw numbers to admit a positive factorization.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1

References

  1. Amorós, J., Bogomolov, F., Katzarkov, L., Pantev, T.: Symplectic Lefschetz fibrations with arbitrary fundamental groups. J. Differ. Geom. 54(3), 489–545 (2000). (With an appendix by Ivan Smith)

    MathSciNet  MATH  Google Scholar 

  2. Bogomolov, F.A., De Oliveira, B..: Stein small deformations of strictly pseudoconvex surfaces. In Birational algebraic geometry. A conference on algebraic geometry in memory of Wei-Liang Chow (1911–1995), Baltimore, MD, USA, April 11–14, 1996, pp 25–41. Providence, RI: American Mathematical Society (1997)

  3. Baker, Kenneth L., Etnyre, John B., Van Horn-Morris, Jeremy: Cabling, contact structures and mapping class monoids. J. Differ. Geom. 90(1), 1–80 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Baykur, R İnanç, Monden, Naoyuki, Van Horn-Morris, Jeremy: Positive factorizations of mapping classes. Algebr. Geom. Topol. 17(3), 1527–1555 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Burns Jr., D., Shnider, S., Wells Jr., R.O.: Deformations of strictly pseudoconvex domains. Invent. Math. 46(3), 237–253 (1978)

    MathSciNet  MATH  Google Scholar 

  6. Catlin, David: A Newlander–Nirenberg theorem for manifolds with boundary. Mich. Math. J. 35(2), 233–240 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Cho, Sanghyun, Choi, Jaeseo: Explicit Sobolev estimates for the Cauchy–Riemann equation on parameters. Bull. Korean Math. Soc. 45(2), 321–338 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Cho, Hong Rae, Cho, Sanghyun, Shon, Kwang Ho: Stability of the estimates for \(\overline{\partial }\)-equation on compact pseudoconvex complex manifolds. Kyushu J. Math. 48(1), 19–34 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back, volume 59 of American Mathematical Society Colloquium Publications. Symplectic Geometry of Affine Complex Manifolds. American Mathematical Society, Providence (2012)

    MATH  Google Scholar 

  10. Colin, Vincent, Honda, Ko: Reeb vector fields and open book decompositions. J. Eur. Math. Soc. 15(2), 443–507 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Cho, Sanghyun: Sobolev estimates of solutions of the \(\overline{\partial }\)-Neumann problem on parameters. Int. J. Math. 13(10), 1027–1042 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Caubel, Clément, Némethi, András, Popescu-Pampu, Patrick: Milnor open books and Milnor fillable contact 3-manifolds. Topology 45(3), 673–689 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Chen, S.-C., Shaw, M.-C.: Partial Differential Equations in Several Complex Variables, Volume 19 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence (2001)

    Google Scholar 

  14. Eisenbud, D., Neumann, W.: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. Annals of Mathematics Studies, vol. 110. Princeton University Press, Princeton (1985)

    MATH  Google Scholar 

  15. Folland, G.B., Kohn, J.J.: The Neumann Problem for the Cauchy–Riemann Complex. Annals of Mathematics Studies, vol. 75. Princeton University Press, Princeton (1972)

    MATH  Google Scholar 

  16. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  17. Gabai, David: Problems in foliations and laminations. Geometric Topology, (Athens, GA, 1993), Volume 2 of AMS/IP Studies in Advanced Mathematics, pp. 1–33. American Mathematical Society, Providence (1997)

    Google Scholar 

  18. Giroux, E.: Géométrie de contact: de la dimension trois vers les dimensions supérieures. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002). Higher Ed. Press, Beijing, pp 405–414 (2002)

  19. Greene, Robert E., Krantz, Steven G.: Deformation of complex structures, estimates for the \(\bar{\partial }\) equation, and stability of the Bergman kernel. Adv. Math. 43(1), 1–86 (1982)

    MathSciNet  MATH  Google Scholar 

  20. Gong, Xianghong, Kim, Kang-Tae: The \(\overline{\partial }\)-equation on variable strictly pseudoconvex domains. Math. Z. 290(1–2), 111–144 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Henkin, G.M.: Integral representation of functions which are holomorphic in strictly pseudoconvex regions, and some applications. Mat. Sb. (N.S.) 78(120), 611–632 (1969)

    MathSciNet  Google Scholar 

  22. Henkin, G.M.: Integral representation of functions in strongly pseudoconvex regions, and applications to the \(\overline{\partial }\)-problem. Mat. Sb. (N.S.) 82(124), 300–308 (1970)

    MathSciNet  Google Scholar 

  23. Honda, Ko, Kazez, William H., Matić, Gordana: Right-veering diffeomorphisms of compact surfaces with boundary. Invent. Math. 169(2), 427–449 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Honda, Ko, Kazez, William H., Matić, Gordana: Right-veering diffeomorphisms of compact surfaces with boundary. II. Geom. Topol. 12(4), 2057–2094 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Henkin, Gennadi M., Leiterer, Jürgen: Global integral formulas for solving the \(\bar{\partial }\)-equation on Stein manifolds. Ann. Pol. Math. 39, 93–116 (1981)

    MathSciNet  MATH  Google Scholar 

  26. Henkin, G., Leiterer, J.: Theory of Functions on Complex Manifolds. Monographs in Mathematics, vol. 79. Birkhäuser Verlag, Basel (1984)

    Google Scholar 

  27. Hörmander, Lars: \(L^{2}\) estimates and existence theorems for the \(\bar{\partial }\) operator. Acta Math. 113, 89–152 (1965)

    MathSciNet  MATH  Google Scholar 

  28. Kerzman, Norberto: Hölder and \(L^{p}\) estimates for solutions of \(\bar{\partial }u=f\) in strongly pseudoconvex domains. Commun. Pure Appl. Math. 24, 301–379 (1971)

    MathSciNet  MATH  Google Scholar 

  29. Kohn, J.J., Nirenberg, L.: Non-coercive boundary value problems. Commun. Pure Appl. Math. 18, 443–492 (1965)

    MathSciNet  MATH  Google Scholar 

  30. Kohn, J.J.: Harmonic integrals on strongly pseudo-convex manifolds. I. Ann. Math. 2(78), 112–148 (1963)

    MathSciNet  MATH  Google Scholar 

  31. Kohn, J.J.: Harmonic integrals on strongly pseudo-convex manifolds. II. Ann. Math. 2(79), 450–472 (1964)

    MathSciNet  MATH  Google Scholar 

  32. Laufer, Henry B.: Versal deformations for two-dimensional pseudoconvex manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7(3), 511–521 (1980)

    MathSciNet  MATH  Google Scholar 

  33. Laufer, Henry B.: Lifting cycles to deformations of two-dimensional pseudoconvex manifolds. Trans. Am. Math. Soc. 266, 183–202 (1981)

    MathSciNet  MATH  Google Scholar 

  34. Loi, Andrea, Piergallini, Riccardo: Compact Stein surfaces with boundary as branched covers of \(B^4\). Invent. Math. 143(2), 325–348 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Milnor, John: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton (1968)

    Google Scholar 

  36. Matsumoto, Y., Montesinos-Amilibia, J.: Pseudo-Periodic Maps and Degeneration of Riemann Surfaces. Lecture Notes in Mathematics, vol. 2030. Springer, Heidelberg (2011)

    MATH  Google Scholar 

  37. Marinescu, George, Yeganefar, Nader: Embeddability of some strongly pseudoconvex CR manifolds. Trans. Am. Math. Soc. 359(10), 4757–4771 (2007)

    MathSciNet  MATH  Google Scholar 

  38. Pichon, Anne: Fibrations sur le cercle et surfaces complexes. Ann. Inst. Fourier (Grenoble) 51(2), 337–374 (2001)

    MathSciNet  MATH  Google Scholar 

  39. Popescu-Pampu, Patrick: On the smoothings of non-normal isolated surface singularities. J. Singul. 12, 164–179 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Siersma, Dirk: A bouquet theorem for the milnor fibre. J. Algebr. Geometry 4, 51–66 (1995)

    MathSciNet  MATH  Google Scholar 

  41. Thurston, William P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. (N.S.) 19(2), 417–431 (1988)

    MathSciNet  MATH  Google Scholar 

  42. Tibǎr, Mihai: Limits of tangents and minimality of complex links. Topology 42(3), 629–639 (2003)

    MathSciNet  MATH  Google Scholar 

  43. Tráng, L.D.: Some remarks on relative monodromy. In: Real and Complex Singularities (Proceedings of Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 397–403 (1977)

  44. Wand, Andy: Factorizations of diffeomorphisms of compact surfaces with boundary. Geom. Topol. 19(5), 2407–2464 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Winters, Gayn B.: On the existence of certain families of curves. Am. J. Math. 96, 215–228 (1974)

    MathSciNet  MATH  Google Scholar 

  46. Wittmann, Johannes: The Banach manifold \(C^k(M, N)\). Differ. Geom. Appl. 63, 166–185 (2019)

    MATH  Google Scholar 

Download references

Acknowledgements

I wish to thank Xavier Gómez-Mont for inspiring conversations. I am also very thankful to Mohammad Jabbari who is an expert in the \({\bar{\partial }}\)-problem and gave me many useful references that lead me to find the Greene and Krantz result that I ended up using. Thanks to Baldur Sigurðsson who read carefully an early version of this manuscript and pointed out a gap in a lemma that was placed instead of current Proposition 4.1. His critics and comments have helped me greatly improve the final manuscript. Finally I thank the very thorough review made by one of the referees whose numerous remarks helped me to improve many parts of the article as well as to fix some proofs that were insufficiently explained or contained gaps in an early version of this article.

Author information

Authors and Affiliations

  1. CIMAT, Guanajuato, Mexico

    Pablo Portilla Cuadrado

Authors
  1. Pablo Portilla Cuadrado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo Portilla Cuadrado.

Ethics declarations

Conflicts of interest

The corresponding author states that there is no conflict of interest.

Data sharing

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Communicated by Thomas Schick.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is supported by CONACYT project wtih No. 286447.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Portilla Cuadrado, P. Positive factorizations of pseudoperiodic homeomorphisms. Math. Ann. 379, 1173–1203 (2021). https://doi.org/10.1007/s00208-020-02110-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-020-02110-5

Mathematics Subject Classification

search

Navigation