Abstract
In this paper we will give an analytic proof of the following finiteness theorem for holomorphic families of Riemann surfaces: FIniteness Theorem OF Families. Let B be a Riemann surface of finite type. Then, there are only finitely many non-isomorphic and locally non-trivial holomorphic families of Riemann surfaces of fixed finite type (g, n) with 2g — 2 + n > 0 over B.
Dedicated to Professor Tadashi Kuroda on his sixtieth birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abikoff, W., Two theorems on totally degenerate Kleinian groups ,Amer. J. Math. 98 (1976), 109–118.
Abikoff, W., “The Real Analytic Theory of Teichmüller Space,” Lecture Notes in Math. 820, Springer-Verlag, 1980.
Arakelov, S. Ju, Families of curves with fixed degenerancies ,Math. USSR Izvestija 5 (1971), 1277–1302.
Bers, L., An extremal problem for quasiconformal mappings and a theorem by Thurston ,Acta Math. 141 (1978), 73–98.
Bers, L., Finite dimensional Teichmüller spaces and generalizations ,Bull. Amer. Math. Soc. 5 (1981), 131–172.
Bers, L., On iterations of hyperbolic transformations of Teichmüller spaces ,Amer. J. of Math. 105 (1983), 1–11.
Faltings, G., Arakelov’s theorem for Abelian varieties ,Invent. Math. 73 (1983), 337–347.
Faltings, G., Endlichkeitssäte für abelshe Varietäten über Zahlkörpern ,Invent. Math. 73 (1983), 349–366.
Grauert, H., Mordells Vermutung über Punkte auf algebraischen Kurven und Funktionenkörper ,Publ. Math. I.H.E.S. 25 (1965), 131–149.
Kobayashi, S., “Hyperbolic Manifolds and Holomorphic Mappings,” Marcel Dekker, Inc., New York, 1970.
Mazur, B., Arithmetic on curves ,Bull. (new series) of Amer. Math. Soc. 14 (1986), 207–259.
Miwa, M., On Mordell’s conjecture for algebraic curves over function fields ,J. Math. Soc. Japan 18 (1966), 182–188.
Mumford, D., “Curves and Their Jacobians,” The University of Michigan Press, Ann Arbor, 1975.
Noguchi, J., A higher dimensional analogue of Mordell’s conjecture over function fields ,Math. Ann. 258 (1981), 207–212.
Noguchi, J., Hyperbolic fibre spaces and Mordell’s conjecture over function fields ,Publ. RIMS, Kyoto Univ. 21 (1985), 27–46.
Parshin, A.N., Algebraic curves over function fields I ,Math. USSR Izvestija 2 (1968), 1145–1170.
Samuel, S., La conjecture de Mordell pour les corp fe fonctions ,Séminaire Bourbaki 17e année, 1964/65, n 287.
Shafarevich, I.R., Algebraic number fields ,Amer. Math. Soc. Transl (2) 31 (1963), 25–39.
Shiga, H., On analytic and geometric properties of Teichmüller spaces ,J. Math. Kyoto Univ. 24 (1984), 441–452.
Szpiro, L., Sur le théorème de rigidité de Parsin et Arakelov ,Astérisque 64 (1979), 169–202.
Tsuji, M., “Potential Theory in Modern Function Theory,” Chelsea Publishing Company, New York, 1975.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer-Verlag New York Inc.
About this paper
Cite this paper
Imayoshi, Y., Shiga, H. (1988). A finiteness theorem for holomorphic families of Riemann surfaces. In: Drasin, D., Earle, C.J., Gehring, F.W., Kra, I., Marden, A. (eds) Holomorphic Functions and Moduli II. Mathematical Sciences Research Institute Publications, vol 11. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9611-6_15
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9611-6_15
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-9613-0
Online ISBN: 978-1-4613-9611-6
eBook Packages: Springer Book Archive