An analytic proof of Severi’s theorem. (English) Zbl 0585.32025

Imayoshi, Yoichi

Complex Variables, Theory Appl. 2, 151-155 (1983).

A classical theorem of Severi states that a fixed compact Riemann surface R may be mapped holomorphically onto only finitely many Riemann surfaces of genus greater than one. The pesent paper gives an analytic proof of a more general result. It is proved that: If R is a hyperbolic Riemann surface of finite conformal type then for at most finitely many hyperbolic surfaces S are there non-constant holomorphic mappings from R into S. The proof is a clever application of deformation-theoretic techniques.

Reviewer: W.Abikoff

MSC:

32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F10	Compact Riemann surfaces and uniformization
14E05	Rational and birational maps

Keywords:

theorem of Severi; hyperbolic Riemann surface of finite conformal type