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Transactions of the American Mathematical Society

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

 
 

 

Upper bound for distortion of capacity under conformal mapping


Author: Robert E. Thurman
Journal: Trans. Amer. Math. Soc. 346 (1994), 605-616
MSC: Primary 30C85; Secondary 30C20, 30C70
DOI: https://doi.org/10.1090/S0002-9947-1994-1270669-2
MathSciNet review: 1270669
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Abstract: For a finitely-connected domain $ \Omega $ containing $ \infty $, with boundary $ \Gamma $, the logarithmic capacity $ d(\Gamma )$ is invariant under normalized conformal maps of $ \Omega $. But the capacity of a subset $ A \subset \Gamma $ will likely be distorted by such a map. Duren and Schiffer showed that the sharp lower bound for the distortion of the capacity of such a set is the so-called "Robin capacity" of the set $ A$. We present here the sharp upper bound for the distortion, in terms of conformal invariants of $ \Omega $: the harmonic measures of the boundary components of $ \Omega $ and the periods of their harmonic conjugates (the Riemann matrix), and the capacity of $ \Gamma $. In particular, the upper bound depends only on knowing which components of $ \Gamma $ contain parts of $ A$, not on the specific distribution of $ A$. An extremal configuration is described explicitly for a special case.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1270669-2
Keywords: Conformal mapping, logarithmic capacity, extremal problems, variational methods, theta functions, Green's function, harmonic measure, multiply-connected domains
Article copyright: © Copyright 1994 American Mathematical Society

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