Guiding Isotopies and Holomorphic Motions

TL;DR

This paper establishes an isotopy principle for holomorphic motions on Riemann surfaces, providing a method to extend motions holomorphically guided by quasiconformal isotopies, and offers a new proof of Slodkowski's theorem.

Contribution

It introduces a new extendability theorem for holomorphic motions guided by quasiconformal isotopies on Riemann surfaces.

Findings

Holomorphic motions can be extended holomorphically guided by quasiconformal isotopies.

A canonical method replaces quasiconformal motions with holomorphic motions for additional points.

Provides a new proof of Slodkowski's theorem for the Riemann sphere.

Abstract

We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion of a finite subset $E$ of a Riemann surface $Y$ parameterized by a point $t$ in a pointed hyperbolic surface $(X, t_{0})$. If a holomorphic motion from $E$ to $E_{t}$ in $Y$ has a guiding quasiconformal isotopy, then there is a holomorphic extension to any new point $p$ in $Y-E$ that follows the guiding isotopy. The proof gives a canonical way to replace a quasiconformal motion of the $(n+1)^{th}$ point by a holomorphic motion while leaving unchanged the given holomorphic motion of the first $n$ points. In particular, our main result gives a new proof of Slodkowski's theorem which concerns the special case when the parameter space is the open unit disk with base point $0$ and the dynamical space $Y$ is the Riemann sphere.