On extendibility of a map induced by Bers isomorphism

Abstract: Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus {\hat{z}_0 }$. Then we have the composed map $\varphi\circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $\varphi: F(S) \rightarrow T(\dot{S})$, where $F(S)$ is the Bers fiber space of $S$, $T(X)$ is the Teichm\"uller space of $X$ and $U$ is the upper half-plane. The purpose of this paper is to show the map $\varphi\circ r:T(S)\times U \rightarrow T(\dot{S})$. has a continuous extension to some subset of the boundary $T(S) \times \partial U$.