Bell representations of finitely connected planar domains

Jeong, Moonja; Taniguchi, Masahiko

doi:10.1090/S0002-9939-02-06823-5

Bell representations of finitely connected planar domains
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by Moonja Jeong and Masahiko Taniguchi PDF

Proc. Amer. Math. Soc. 131 (2003), 2325-2328 Request permission

Abstract:

In this paper, we solve a conjecture of S. Bell (1992) affirmatively. Actually, we prove that every non-degenerate $n$-connected planar domain $\Omega$, where $n>1$ is representable as $\Omega = \{|f|<1\}$ with a suitable rational function $f$ of degree $n$. This result is considered as a natural generalization of the classical Riemann mapping theorem for simply connected planar domains.

References

Steven R. Bell, The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1228442
Steven R. Bell, Finitely generated function fields and complexity in potential theory in the plane, Duke Math. J. 98 (1999), no. 1, 187–207. MR 1687563, DOI 10.1215/S0012-7094-99-09805-8
Steven R. Bell, A Riemann surface attached to domains in the plane and complexity in potential theory, Houston J. Math. 26 (2000), no. 2, 277–297. MR 1814239
Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745
Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
S. M. Natanzon, Hurwitz spaces, Topics on Riemann surfaces and Fuchsian Groups, LMS Lecture Notes, 287 (2001), 165–177.