It is likely that this is a dumb question.
$D\subset\hat{C}$ is a domain of riemann sphere $\hat{C}$(i.e. it is connected and open). Then $Aut(D)\cong H,H\subset Aut(\hat{C})$ where $H$ is a subgroup of $Aut(\hat{C})$.
Q1. It is obviously true for $D=\Delta(1)$ unit disk and any simply connected domain is diffeo to $\Delta(1)$. Hence it is an restriction of some automorphism of $Aut(\hat{C})$. The question is whether it holds in general, say $D$ is no longer simply connected but connected? In particular, given $f\in Aut(D)$, I can extend it uniquely to $Aut(\hat{C})$.
Q2. It seems that one cannot drop the condition that $D$ is connected. If $U_1,U_2$ are 2 disconnnected open set. So I expect $Aut(U_1\cup U_2)=Aut(U_1)\times Aut(U_2)$ and this might not be embedded as a subgroup of $Aut(\hat{C})$.
The reason I am asking this question is that for any finite extension $L$ over $Q$, the morphism $f:L\to C$ can be extended to $\tilde{f}:C\to C$. I am wonder whether this holds in analogy.