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Microsoft Patchday Summary March 13, 2018 · Security Updates for Windows.. Aiseesoft PDF to Word Converter 3.3.26 Patch [CracksNow] free downloadQ:

Can a child of a conformal factor be conformal?

Let $Omega subset mathbb R^2$ and let $fin C^2$ be a conformal map, mapping $Omega$ onto itself.
I would like to know whether, given any $Omega_1 subset Omega$,
$$
f^{ -1}(Omega_1) subset Omega
$$
is also conformal in the sense that
$$
f:f^{ -1}(Omega_1) to Omega_1
$$
is conformal?
If not, is it true that $f$ is the restriction of a conformal map on some larger domain?

A:

Your problem has a positive answer.
Fix a conformal coordinate system $z=(u,v)$ on $Omega$. Then, a priori, there exists $R,epsilon>0$ so that $f(B(0,R))subsetOmega^epsilon$. In fact, fixing $z_0in B(0,R)$, we have that $f_*
u_z=
u_{f(z)}$ for all $zin B(z_0,epsilon)$. So $f$ is an orientation-preserving diffeomorphism on $B(z_0,epsilon)$ (and hence on $Omega$).
Now suppose, for some $z_0in B(0,R)$, that $f_*
u_z$ is $C^1$ in some neighborhood $UsubsetOmega^epsilon$. Then we have that
$$

abla_z fcdot frac{partial f_*
u_z}{partial u} =
abla_z fcdot frac{partial f_*
u_z}{partial v} =
abla_z fcdot f_*
u_z =1
$$
so that

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