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On the number of holomorphic mappings between Riemann surfaces of finite analytic type

Part of: Curves Holomorphic mappings and correspondences

Published online by Cambridge University Press:  20 June 2011

Yoichi Imayoshi ,
Manabu Ito  and
Hiroshi Yamamoto
Yoichi Imayoshi
Affiliation:
Department of Mathematics, Osaka City University, Sugimotocyo, Sumiyoshi-ku, Osaka 558-0022, Japan (imayoshi@sci.osaka-cu.ac.jp)
Manabu Ito
Affiliation:
10-20-101, Hirano-kita 1-chome, Hirano-ku, Osaka 547-0041, Japan (cbj89070@pop02.odn.ne.jp)
Hiroshi Yamamoto
Affiliation:
Department of Integrated Arts and Science, Okinawa National College of Technology, 905 Henoko, Nago City, Okinawa 905-2192, Japan (yamamoto@okinawa-ct.ac.jp)
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Abstract

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The set of non-constant holomorphic mappings between two given compact Riemann surfaces of genus greater than 1 is always finite. This classical statement was made by de Franchis. Furthermore, bounds on the cardinality of the set depending only on the genera of the surfaces have been obtained by a number of mathematicians. The analysis is carried over in this paper to the case of Riemann surfaces of finite analytic type (i.e. compact Riemann surfaces minus a finite set of points) so that the finiteness result, together with a crude but explicit bound depending only on the topological data, may be extended for the number of holomorphic mappings between such surfaces.

Keywords

holomorphic mappingRiemann surface of finite analytic typeEuler–Poincaré characteristic

MSC classification

Secondary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions 14H30: Coverings, fundamental group
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 711 - 730
Copyright
Copyright © Edinburgh Mathematical Society 2011

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