## On conformal maps of infinitely connected Dirichlet regions

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- by V. C. Williams PDF
- Trans. Amer. Math. Soc.
**155**(1971), 427-453 Request permission

## 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.## References

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## Additional Information

- © 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