Jordan curve and Conformal maps
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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?
complex-analysis riemann-surfaces conformal-geometry
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
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add a comment |
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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?
complex-analysis riemann-surfaces conformal-geometry
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
$endgroup$
add a comment |
$begingroup$
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?
complex-analysis riemann-surfaces conformal-geometry
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
$endgroup$
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?
complex-analysis riemann-surfaces conformal-geometry
complex-analysis riemann-surfaces conformal-geometry
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
asked Dec 22 '18 at 15:29
Claudio DelfinoClaudio Delfino
63
63
add a comment |
add a comment |
1 Answer
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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 $f_2circ f_1^{-1}:overline D_1tooverline D_2$.
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
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$begingroup$
I can' t find this theorem in any book
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– Claudio Delfino
Dec 22 '18 at 21:56
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@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.
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– David C. Ullrich
Dec 23 '18 at 14:37
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 $f_2circ f_1^{-1}:overline D_1tooverline D_2$.
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $f_2circ f_1^{-1}:overline D_1tooverline D_2$.
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $f_2circ f_1^{-1}:overline D_1tooverline D_2$.
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
$endgroup$
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 $f_2circ f_1^{-1}:overline D_1tooverline D_2$.
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
edited Dec 22 '18 at 18:55
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
answered Dec 22 '18 at 18:49
David C. UllrichDavid C. Ullrich
61.8k44095
61.8k44095
$begingroup$
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
add a comment |
$begingroup$
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
$begingroup$
I can' t find this theorem in any book
$endgroup$
– Claudio Delfino
Dec 22 '18 at 21:56
$begingroup$
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
$begingroup$
@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
add a comment |
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