Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected?
In particular, I am curious about the following question: given a doubly connected smooth $\Omega\subset \mathbb{C}$, does there exists a continuous family of biholomorphic maps $f_t \colon B_1 -\overline{B}_{\beta} \to \mathbb{C}$, $0\leq t \leq 1$, such that
- $f_0(z)=z$,
- $f_1$ maps $B_1 - \overline{B}_{\beta}$ onto $\Omega$,
where $\beta$ is picked so that $B_1 - \overline{B}_{\beta}$ is biholomorphic to $\Omega$?