Abstract
We introduce two basic notions, ‘transboundary extremal length’ and ‘fat sets’, and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short proof is given to Koebe's conjecture in the countable case: every planar domain with countably many boundary components is conformally equivalent to a circle domain. This theorem is further generalized in two direction. We show that the same statement is true for a wide class of domains with uncountably many boundary components, in particular for domains bounded byK-quasicircles and points. Moreover, these domains admit more general uniformizations. For example, every circle domain is conformally equivalent to a domain whose complementary components are heart-shapes and points.
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