Computable Riemann Surfaces

Authors:
Robert Rettinger
Affiliations:
FernUniversität Hagen, LG Komplexität und Algorithmen, Universitätsstrasse 1, D-58095 Hagen,
Venue:
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Year:
2007

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Abstract

In this paper we introduce computable and time bounded Riemann surfaces, based on the classical abstract definition by charts. Building upon this definition we discuss computable versions of several classical results, such as the existence of complete continuations of holomorphic functions, universal coverings and the uniformization theorem (for some cases).Though we state most of our results for computable surfaces, many of them can also be transformed to a uniform version, i.e. based on representations of the class of Riemann surfaces (modulo conformal equivalence).