Abstract:Let $D$ be a plane region of arbitrary connectivity $( > 1)$ for which the Dirichlet problem is solvable. There exists a conformal map of $D$ onto a region bounded by two level loci of $H$, a nontrivial harmonic measure. $H$ is essentially the difference of two logarithmic potentials. The two measures involved are mutually singular probability measures. Further properties of these measures, and of $H$, are derived. The special case in which $D$ is of connectivity 2 is the classical theorem which states that an annular region is conformally equivalent to a region bounded by two circles. The case in which $D$ is of finite connectivity was treated by J. L. Walsh in 1956. A similar generalization of the Riemann mapping theorem is also established. Finally, converses of the above results are also valid.
- Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
- C. Carathéodory, Conformal representation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 28, Cambridge, at the University Press, 1952. 2d ed. MR 0046435
- H. S. M. Coxeter, Introduction to geometry, John Wiley & Sons, Inc., New York-London, 1961. MR 0123930
- Helmut Grunsky, Über konforme Abbildungen, die gewisse Gebietsfunktionen in elementare Funktionen transformieren. I, Math. Z. 67 (1957), 129–132 (German). MR 88547, DOI 10.1007/BF01258849
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869 F. Hausdorff, Mengenlehre, de Gruyter, Berlin, 1937; English transl., Chelsea, New York, 1957. MR 19, 111.
- James A. Jenkins, On a canonical conformal mapping of J. L. Walsh, Trans. Amer. Math. Soc. 88 (1958), 207–213. MR 88549, DOI 10.1090/S0002-9947-1958-0088549-5
- H. J. Landau, On canonical conformal maps of multiply connected domains, Trans. Amer. Math. Soc. 99 (1961), 1–20. MR 121474, DOI 10.1090/S0002-9947-1961-0121474-X R. L. Moore, Concerning the separation of point sets by curves, Proc. Nat. Acad. Sci. U.S.A. 11 (1925), 469-476.
- R. L. Moore, Foundations of point set theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
- J. L. Walsh, On the conformal mapping of multiply connected regions, Trans. Amer. Math. Soc. 82 (1956), 128–146. MR 80727, DOI 10.1090/S0002-9947-1956-0080727-2
- J. L. Walsh and H. J. Landau, On canonical conformal maps of multiply connected regions, Trans. Amer. Math. Soc. 93 (1959), 81–96. MR 160884, DOI 10.1090/S0002-9947-1959-0160884-2
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 155 (1971), 427-453
- MSC: Primary 30.40
- DOI: https://doi.org/10.1090/S0002-9947-1971-0280698-8
- MathSciNet review: 0280698