Let $\mathbb D$ be the unitary open disk, $D$ a bounded domain(open and connected) with boundary a jordan curve and $f$ a conformal map from $\mathbb D$ to $D$, is it true that we can extend $f$ as an homeomorphism from $\overline{\mathbb D}$ to $\overline{D}$? and if it is true, does it hold for two bounded domains with boundary a jordan curve?


Yes to the first question - this is a theorem of Caratheodory.

And this implies a yes to the second question: If $f_j:\overline{\Bbb D}\to \overline D_j$ for $j=1,2$ are as above then $f_2\circ f_1^{-1}:\overline D_1\to\overline D_2$.

  • $\begingroup$ I can' t find this theorem in any book $\endgroup$ – Claudio Delfino Dec 22 '18 at 21:56
  • $\begingroup$ @ClaudioDelfino There's what appears to be a fairly complete proof on the Wikipedia page. A weaker version of the result is in Rudin Real and Complex Analysis, Theorem 14.18; in the comments in section 14.20 he essentially states that the version above is true, no proof. $\endgroup$ – David C. Ullrich Dec 23 '18 at 14:37

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