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

Jiang, Yunping; Mitra, Sudeb

doi:10.1090/ecgd/329

Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions
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by Yunping Jiang and Sudeb Mitra

Conform. Geom. Dyn. 22 (2018), 333-344

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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