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Transactions of the American Mathematical Society

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

 
 

 

Isometries for the modulus metric are quasiconformal mappings


Authors: Dimitrios Betsakos and Stamatis Pouliasis
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 30C62, 30C65; Secondary 30C85, 30F45
DOI: https://doi.org/10.1090/tran/7712
Published electronically: November 21, 2018
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Abstract: For a domain $ D$ in $ \bar {\mathbb{R}}^n$, the modulus metric is defined by $ \mu _D(x,y)=\inf _\gamma {\rm cap}(D,\gamma )$, where the infimum is taken over all curves $ \gamma $ in $ D$ joining $ x$ to $ y$, and ``cap" denotes the conformal capacity of the condensers. It has been conjectured by J. Ferrand, G. J. Martin, and M. Vuorinen that isometries in the modulus metric are conformal mappings. We prove the conjecture when $ n=2$. In higher dimensions, we prove that isometries are quasiconformal mappings.


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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: betsakos@math.auth.gr

Stamatis Pouliasis
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409
Email: stamatis.pouliasis@ttu.edu

DOI: https://doi.org/10.1090/tran/7712
Keywords: Modulus metric, conformal mapping, quasiconformal mapping, condenser capacity, reduced conformal modulus, extremal length.
Received by editor(s): January 31, 2018
Received by editor(s) in revised form: April 18, 2018, and July 1, 2018
Published electronically: November 21, 2018
Article copyright: © Copyright 2018 American Mathematical Society