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Transboundary extremal length

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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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Authors and Affiliations

  1. Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Oded Schramm

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  1. Oded Schramm
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Incumbent of the William Z. and Eda Bess Novick Career Development Chair. Supported by NSF grant DMS-9112150.

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Schramm, O. Transboundary extremal length. J. Anal. Math. 66, 307–329 (1995). https://doi.org/10.1007/BF02788827

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  • DOI: https://doi.org/10.1007/BF02788827

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