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Conformal Geometry and Dynamics

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Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions


Authors: Yunping Jiang and Sudeb Mitra
Journal: Conform. Geom. Dyn. 22 (2018), 333-344
MSC (2010): Primary 32G15; Secondary 30C62, 30F60, 30F99
DOI: https://doi.org/10.1090/ecgd/329
Published electronically: December 12, 2018
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Abstract: We study monodromy of holomorphic motions and show the equivalence of triviality of monodromy of holomorphic motions and extensions of holomorphic motions to continuous motions of the Riemann sphere. We also study liftings of holomorphic maps into certain Teichmüller spaces. We use this ``lifting property'' to prove that, under the condition of trivial monodromy, any holomorphic motion of a closed set in the Riemann sphere, over a hyperbolic Riemann surface, can be extended to a holomorphic motion of the sphere, over the same parameter space. We conclude that this extension can be done in a conformally natural way.


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Yunping Jiang
Affiliation: Department of Mathematics, Queens College of the City University of New York, New York; and Department of Mathematics, The Graduate Center, CUNY, New York, New York
Email: yunping.jiang@qc.cuny.edu

Sudeb Mitra
Affiliation: Department of Mathematics, Queens College of the City University of New York, New York; and Department of Mathematics, The Graduate Center, CUNY, New York, New York
Email: sudeb.mitra@qc.cuny.edu

DOI: https://doi.org/10.1090/ecgd/329
Keywords: Teichm\"uller spaces, holomorphic maps, universal holomorphic motions
Received by editor(s): August 15, 2017
Received by editor(s) in revised form: September 10, 2018
Published electronically: December 12, 2018
Additional Notes: The first author was partially supported by an NSF grant, a collaboration grant from the Simons Foundation (grant number 523341), and a grant from NSFC (grant number 11571122).
Both authors were partially supported by PSC-CUNY grants.
Dedicated: In memory of Professor Clifford J. Earle
Article copyright: © Copyright 2018 American Mathematical Society

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