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

TL;DR

This paper investigates the relationship between monodromy and holomorphic motions, establishing conditions under which such motions can be extended holomorphically in a conformally natural manner, with implications for Teichmüller theory.

Contribution

It proves the equivalence of trivial monodromy and extendability of holomorphic motions, and demonstrates extension results using liftings into Teichmüller spaces.

Findings

Trivial monodromy is equivalent to extendability of holomorphic motions.

Holomorphic motions over hyperbolic Riemann surfaces can be extended to the sphere.

Extensions can be achieved in a conformally natural way.

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 Teichm\"uller 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.