Skip to main content
SpringerLink
Log in

Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus \(g\ge 2\), which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor, and we establish a partial converse for hyperelliptic structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Baba, S.: Complex projective structures with schottky holonomy. Geom. Funct. Anal. 22(2), 267–310 (2012)

    Article  MathSciNet  Google Scholar 

  2. Baba, S.: \(2\pi \)-grafting and complex projective structures. I. Geom. Topol. 19(6), 3233–3287 (2015)

    Article  MathSciNet  Google Scholar 

  3. Baba, S.: \(2\pi \)-grafting and complex projective structures with generic holonomy. Geom. Funct. Anal. 27(5), 1017–1069 (2017)

    Article  MathSciNet  Google Scholar 

  4. Baba, S., Gupta, S.: Holonomy map fibers of \({\mathbb{C}} P ^1\)-structures in moduli space. J. Topol. 8(3), 691–710 (2015)

    Article  MathSciNet  Google Scholar 

  5. Biswas, I., Dumitrescu, S., Heller, S.: Irreducible flat \(\operatorname{SL}\)(2,\({\mathbb{R}}\))-connections on the trivial holomorphic bundle. J. Math. Pure, Appl (2020) ((in press))

  6. Calsamiglia, G., Deroin, B., Francaviglia, S.: Branched projective structures with Fuchsian holonomy. Geom. Topol. 18(1), 379–446 (2014)

    Article  MathSciNet  Google Scholar 

  7. Calsamiglia, G., Deroin, B., Francaviglia, S.: The oriented graph of multi-graftings in the Fuchsian case. Publ. Mat. 58(1), 31–46 (2014)

    Article  MathSciNet  Google Scholar 

  8. Calsamiglia, G., Deroin, B., Heu, V., Loray, F.: The Riemann–Hilbert mapping for \({\mathfrak{sl}}_2\) systems over genus two curves. Bull. Soc. Math. France 147(1), 159–195 (2019)

    Article  MathSciNet  Google Scholar 

  9. Dumas, D.: Complex projective structures. In: Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, pp. 455–508. Eur. Math. Soc., Zürich (2009)

  10. Faraco, G., Ruffoni, L.: Complex projective structures with maximal number of Möbius transformations. Math. Nachr. 292(6), 1260–1270 (2019)

    Article  MathSciNet  Google Scholar 

  11. Gallo, D., Kapovich, M., Marden, A.: The monodromy groups of Schwarzian equations on closed Riemann surfaces. Ann. Math. (2) 151(2), 625–704 (2000)

    Article  MathSciNet  Google Scholar 

  12. Goldman, W.M.: Projective structures with Fuchsian holonomy. J. Differ. Geom. 25(3), 297–326 (1987)

    Article  MathSciNet  Google Scholar 

  13. Hejhal, D.A.: Monodromy groups and linearly polymorphic functions. Acta Math. 135(1), 1–55 (1975)

    Article  MathSciNet  Google Scholar 

  14. Imayoshi, Y., Taniguchi, M.: An introduction to Teichmüller spaces. Springer-Verlag, Tokyo (1992). (Translated and revised from the Japanese by the authors)

  15. Mandelbaum, R.: Branched structures on Riemann surfaces. Trans. Am. Math. Soc. 163, 261–275 (1972)

    Article  MathSciNet  Google Scholar 

  16. Mandelbaum, R.: Branched structures and affine and projective bundles on Riemann surfaces. Trans. Am. Math. Soc. 183, 37–58 (1973)

    Article  MathSciNet  Google Scholar 

  17. Maskit, B.: On a class of Kleinian groups. Ann. Acad. Sci. Fenn. Ser. 442, 8 (1969)

    MathSciNet  MATH  Google Scholar 

  18. Nag, S.: Schiffer variation of complex structure and coordinates for Teichmüller spaces. Proc. Indian Acad. Sci. Math. Sci. 94(2–3), 111–122 (1985)

    Article  MathSciNet  Google Scholar 

  19. Poincaré, H.: Sur les groupes des équations linéaires. Acta Math. 4(1), 201–312 (1884)

    Article  MathSciNet  Google Scholar 

  20. Ruffoni, L.: Bubbling complex projective structures with quasi-fuchsian holonomy. J. Topol, Anal (2019). ((in press))

  21. Ruffoni, L.: Multi(de)grafting quasi-Fuchsian complex projective structures via bubbles. Differe. Geom. Appl. 64, 158–173 (2019)

    Article  MathSciNet  Google Scholar 

  22. Tanigawa, H.: Divergence of projective structures and lengths of measured laminations. Duke Math. J. 98(2), 209–215 (1999)

    Article  MathSciNet  Google Scholar 

  23. Thurston, W.P.: Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton, NJ (1997). Edited by Silvio Levy

Download references

Acknowledgements

We would like to thank Bertrand Deroin and Gabriele Mondello for many useful conversations. This work has been partially supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 777822 (“Geometric and Harmonic Analysis with Interdisciplinary Applications”).

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Stefano Francaviglia

  2. Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL, 32306-4510, USA

    Lorenzo Ruffoni

Authors
  1. Stefano Francaviglia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lorenzo Ruffoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lorenzo Ruffoni.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Francaviglia, S., Ruffoni, L. Local deformations of branched projective structures: Schiffer variations and the Teichmüller map. Geom Dedicata 214, 21–48 (2021). https://doi.org/10.1007/s10711-021-00601-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-021-00601-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation