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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

  1. $f_0(z)=z$,
  2. $f_1$ maps $B_1 - \overline{B}_{\beta}$ onto $\Omega$,

where $\beta$ is picked so that $B_1 - \overline{B}_{\beta}$ is biholomorphic to $\Omega$?

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  • $\begingroup$ Not "modulus of continuity" but "conformal modulus". What topology are you using to define continuity? Uniform convergence on compacts? $\endgroup$ Commented 2 days ago
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    $\begingroup$ See the answer here. If you use the topology of uniform convergence on compacts, the Alex answered your question. $\endgroup$ Commented 2 days ago
  • $\begingroup$ Do you understand that modulus of continuity and conformal modulus are totally different concepts? $\endgroup$ Commented 2 days ago
  • $\begingroup$ @MoisheKohan Sorry, can you explain the difference? I only know modulus of continuity from Rudin. And yes, Uniform convergence on compacts. $\endgroup$
    – Jinyang wu
    Commented 2 days ago
  • $\begingroup$ Google "conformal modulus". $\endgroup$ Commented 2 days ago

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