Abstract
In this seminal paper, Schiffer introduced an important conformal invariant for multiply connected plane domains, the span. The definition is in terms of conformal mappings onto canonical slit domains, and there is a complementary characterization (almost literally), in terms of an extremal problem for area under a conformal mapping. While the foundational results on conformal mapping of multiply connected domains concern the existence and uniqueness of mappings onto various canonical domains, Schiffer’s paper is an early example of using the mappings as a tool in geometric function theory. Moreover, his study of the extremal problem is clearly influenced by his development of variational methods.
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Notes
- 1.
In [AB] most results are given for z 0 a finite point in D, unlike Schiffer’s original normalization, but this is not essential (nor was it essential for Schiffer to use z 0 = ∞).
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Osgood, B. (2013). [14] The span of multiply connected domains. In: Duren, P., Zalcman, L. (eds) Menahem Max Schiffer: Selected Papers Volume 1. Contemporary Mathematicians. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-8085-5_17
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DOI: https://doi.org/10.1007/978-0-8176-8085-5_17
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