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Holomorphic mappings between compact Riemann surfaces

Part of: Riemann surfaces Holomorphic mappings and correspondences

Published online by Cambridge University Press:  02 February 2009

Manabu Ito  and
Hiroshi Yamamoto
Manabu Ito
Affiliation:
10-20-101, Hirano-kita 1 chome, Hirano-ku, Osaka 547-0041, Japan ([email protected])
Hiroshi Yamamoto
Affiliation:
Department of Integrated Arts and Science, Okinawa National College of Technology, 905 Henoko, Nago City, Okinawa 905-2192, Japan ([email protected])
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Abstract

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There are only finitely many non-constant holomorphic mappings between two fixed compact Riemann surfaces of genus greater than 1. This result goes under the name of the de Franchis theorem. Having seen that the set of such holomorphic mappings is finite, we naturally want to obtain a bound on its cardinality. It has been known for some time that there exist various bounds depending only on the genera of the surfaces. Here we obtain ‘better’ bounds of the above type, using arguments based on the rigidity of holomorphic mappings and the hyperbolic geometry of surfaces.

Keywords

compact Riemann surfaceholomorphic mappingSchwartz lemmarigidity theorems for holomorphic mappingscollar theoremlength-controlled pants decompositions

MSC classification

Secondary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions 30F10: Compact Riemann surfaces and uniformization 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 52 , Issue 1 , February 2009 , pp. 109 - 126
Copyright
Copyright © Edinburgh Mathematical Society 2009