On the multiplicity of the image of simple closed curves via holomorphic maps between compact Riemann surfaces

Kodai Mathematical Journal

Hiroshi Yamamoto¹ 

[1] DEPARTMENT OF MATHEMATICS OSAKA CITY UNIVERSITY

关键词: Compact Riemann surface;    holomorphic function;    simple closed curve;    multiplicity of closed curves;   

DOI  :  10.2996/kmj/1050496649

学科分类：数学（综合）

来源: Tokyo Institute of Technology, Department of Mathematics

【 摘 要 】

References(5)Every non-trivial closed curve C on a compact Riemann surface R is freely homotopic to the r-fold iterate C0r of some primitive closed geodesic C0 on R. We call r the multiplicity of C, and denote it by NR(C). Let f be a non-constant holomorphic map of a compact Riemann surface R1 of genus g1 onto another compact Riemann surface R2 of genus g2 with g1≥g2>1, and C a simple closed geodesic of hyperbolic length lR1(C) on R1. In this paper, we give an upper bound for NR2(f(C)) depending only on g1, g2 and lR1(C).

【 授权许可】

Unknown