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Conformal Geometry and Dynamics

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Analytic capacity and holomorphic motions


Authors: Stamatis Pouliasis, Thomas Ransford and Malik Younsi
Journal: Conform. Geom. Dyn. 23 (2019), 130-134
MSC (2010): Primary 30C85; Secondary 31A15, 37F99
DOI: https://doi.org/10.1090/ecgd/336
Published electronically: July 2, 2019
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Abstract: We study the behavior of the analytic capacity of a compact set under deformations obtained by families of conformal maps depending holomorphically on the complex parameter. We show that, under those deformations, the logarithm of the analytic capacity varies harmonically. We also show that the hypotheses in this result cannot be substantially weakened.


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Additional Information

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

Thomas Ransford
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec (Québec), G1V 0A6, Canada
Email: thomas.ransford@mat.ulaval.ca

Malik Younsi
Affiliation: Department of Mathematics, University of Hawaii Manoa, Honolulu, Hawaii 96822
Email: malik.younsi@gmail.com

DOI: https://doi.org/10.1090/ecgd/336
Keywords: Analytic capacity, holomorphic motion, harmonic function
Received by editor(s): September 10, 2018
Published electronically: July 2, 2019
Additional Notes: The second author was supported by grants from NSERC and the Canada Research Chairs program.
The third author was supported by NSF Grant DMS-1758295
Article copyright: © Copyright 2019 American Mathematical Society