Proceedings of the Japan Academy, Series A, Mathematical Sciences

A criterion for holomorphic families of Riemann surfaces to be virtually isomorphic

Hideki Miyachi

Full-text: Open access

Abstract

In this paper, we give a rigidity theorem for holomorphic disks associated to holomorphic families of Riemann surfaces, aiming for studying the geometry (the asymptotic boundary behaviors) of such holomorphic disks. Indeed, our rigidity theorem is described with distributions of associated holomorphic disks and orbits of monodromies at infinity.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 10 (2015), 151-154.

Dates
First available in Project Euclid: 2 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1449080126

Digital Object Identifier
doi:10.3792/pjaa.91.151

Mathematical Reviews number (MathSciNet)
MR3430204

Zentralblatt MATH identifier
1348.30024

Subjects
Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Keywords
Holomorphic family Teichmüller space Teichmüller modular group Teichmüller distance

Citation

Miyachi, Hideki. A criterion for holomorphic families of Riemann surfaces to be virtually isomorphic. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 10, 151--154. doi:10.3792/pjaa.91.151. https://projecteuclid.org/euclid.pja/1449080126


Export citation

References

  • S. J. Arakelov, Families of algebraic curves with fixed degeneracies, Math. USSR Izv. 5 (1971), no. 6, 1277–1302.
  • H. Grauert, Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 131–149.
  • M. Gromov, Hyperbolic groups, in Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, pp. 75–263.
  • Y. Imayoshi and H. Shiga, A finiteness theorem for holomorphic families of Riemann surfaces, in Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), 207–219, Math. Sci. Res. Inst. Publ., 11, Springer, New York, 1988.
  • Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, translated and revised from the Japanese by the authors, Springer, Tokyo, 1992.
  • H. Miyachi, A rigidity theorem for holomorphic disks in Teichmüller space, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2949–2957.
  • M. Miwa, On Mordell's conjecture for algebraic curves over function fields, J. Math. Soc. Japan 18 (1966), no. 2,182–188.
  • J. Noguchi, Hyperbolic fibre spaces and Mordell's conjecture over function fields, Publ. Res. Inst. Math. Sci. 21 (1985), no. 1, 27–46.