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2012•arXiv (Cornell University)Open access

Asymptotic conformal welding via Loewner-Kufarev evolution

Dmitri Prokhorov

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Abstract

The Loewner-Kufarev evolution produces asymptotics for mappings onto domains close to the unit disk or the exterior of the unit disk. We deduce variational formulae which lead to the asymptotic conformal welding for such domains. The comparison of mappings onto bounded and unbounded components of the Jordan curve establishes an asymptotic connection between driving functions in both versions of the Loewner-Kufarev equation and conformal radii of the two domains.

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The Loewner-Kufarev evolution produces asymptotics for mappings onto domains close to the unit disk or the exterior of the unit disk. We deduce variational formulae which lead to the asymptotic conformal welding for such domains. The comparison of mappings onto bounded and unbounded components of the Jordan curve establishes an asymptotic connection between driving functions in both versions of the Loewner-Kufarev equation and conformal radii of the two domains.

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Available abstract

The Loewner-Kufarev evolution produces asymptotics for mappings onto domains close to the unit disk or the exterior of the unit disk. We deduce variational formulae which lead to the asymptotic conformal welding for such domains. The comparison of mappings onto bounded and unbounded components of the Jordan curve establishes an asymptotic connection between driving functions in both versions of the Loewner-Kufarev equation and conformal radii of the two domains.

Key concepts: Conformal map, Unit disk, Connection (principal bundle), Mathematics, Bounded function, Extremal length, Welding, Unit (ring theory)

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