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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On conformal maps of infinitely connected Dirichlet regions

Author: V. C. Williams
Journal: Trans. Amer. Math. Soc. 155 (1971), 427-453
MSC: Primary 30.40
MathSciNet review: 0280698
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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.

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Keywords: Infinitely connected Dirichlet region, harmonic measure, logarithmic potential, subharmonic function
Article copyright: © Copyright 1971 American Mathematical Society

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