A note on the extendability of holomorphic motions

Abstract

In this paper, we consider conditions under which a holomorphic motion of a closed subset of $\mathbf{Ĉ}$ over a non-simply connected Riemann surface $X$ can be extended to a holomorphic motion of $\mathbf{Ĉ}$ over $X$. We construct examples of non-extendable holomorphic motions which satisfy fairy good topological conditions. The examples are also counter-examples to a claim by Chirka for the extendability of holomorphic motions.