Finiteness Theorems for Holomorphic Maps

摘要: This Part is devoted to a direction of complex analysis which has its roots in the theorem Liouville (Liouville (1844) for doubly periodic functions; Cauchy contemporary formulation) and Picard (1879) on nonexistence nonconstant holomorphic functions f: ℂ→D = {z∈ ℂ: |z| 1. Hurwitz (1893) completed this result by establishing explicit bound # Aut R g ≦84(g — 1); we owe him several other remarkable results maps Riemann surfaces (some these will be set forth § 2 3 Chap. 1). De Franchis (1913) Severi (1926) proved finiteness Hol*(R g1, g2) compact g1→R g2 hypothesis 2> Moreover, they established that, fixed surface number all pairs (f, g2), where genus 2>1 f∈Hol*(R finite (and admits estimates depending only