0
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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 $f_2\circ f_1^{-1}:\overline D_1\to\overline D_2$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.