Conformal mapping is the key object of studying in geometric function theory of one complex variable. In this area, the most important and fundamental result is the Riemann mapping theorem, which states that any proper simply connected domain D on
For polygon areas, Riemann mappings have some special forms, known as the Schwarz-Christoffel formula[1]. In Ref.[2], Thurston gave a geometric approach to the construction of Riemann mappings. Thurston's method was further developed in Ref.[3]. Based on Bishop's methods[4], Cheng[5] studied explicit construction of Riemann mapping, by considering a condition called mappability.
In this note, we give a different explicit construction of iterative approximation of Riemann mappings. Our method is motivated by Koebe's proof of the Riemann mapping theorem, as presented in Ahlfors' classic Ref.[1]. The construction in the present note can be easily realized as a computer program, except the choices of the points wk (see section 2 for details) which will be further studied in forthcoming works.
1 Proof of Riemann mapping theoremIn this section, we present a proof of the Riemann mapping theorem which is due to Koebe (see Ahlfors[1]). Inspired by the idea in this proof, we will give an explicit iterative construction of conformal mappings from simply connected domains onto the unit disc.
Theorem 1.1 Let D⊆
Proof The proof is taken from Ref.[1]. The analytic property encoded by the simply connectedness that will be used in the proof is the existence of square root of any holomorphic function on D without zeros. The uniqueness part follows easily from Schwarz Lemma. We give a proof of the existence of f, which is divided into several steps as follows.
Step 1. After a translation, we may assume 0∉D. Since D is simply connected,
$ {\sigma _ + }(D) \cap {\sigma _ - }(D) = \varnothing . $ |
In fact, if the formula does not hold, then there exist z, w∈D such that σ+(z)=σ-(w), and then z=σ+2(z)=σ-2(w)=w. This implies σ+(z)=σ-(w)=-σ+(z) and z=0. We get a contradiction since 0∉D.
It is also clear that σ+ and σ- are injective, hence we have σ+(D)
For a point a∈σ-(D), we can find r>0 such that
Step 2. We consider the set:
$ \left| {{f^\prime }\left( {{z_0}} \right)} \right| = \mathop {\lim }\limits_{j \to + \infty } \left| {g_j^\prime \left( {{z_0}} \right)} \right| = \lambda > 0. $ |
Hence f is injective by the Hurwitz's lemma. Note that f(D)⊆Δ, we can get f∈
Step 3. The last step is to show that f is surjective. We argue by contradiction. If f is not subjective, there exists a point w0∈Δ\f(D). Set
$ {\psi _{{w_0}}}(z) = \frac{{z - {w_0}}}{{1 - \overline {{w_0}}z }}, $ |
then ψw0∈Aut(Δ) and 0≠ψw0
$ \tilde f = {\psi _{h\left( { - {w_0}} \right)}} \circ h \circ {\psi _{{w_0}}} \in \widetilde {\cal F}. $ |
A straightforward calculation shows:
$ \left| {{{\tilde f}^\prime }\left( {{z_0}} \right)} \right| = \frac{{1 + \left| {{w_0}} \right|}}{{2\sqrt {\mid {w_0}} \mid }}\left| {{f^\prime }\left( {{z_0}} \right)} \right| > \left| {{f^\prime }\left( {{z_0}} \right)} \right| = \lambda . $ |
This leads to a contradiction by the definition of λ.
After a gyration, we have f'(z0)>0.
The mapping in the above theorem is called a Riemann mapping.
2 Iterative construction of Riemann mappingMotived by the proof of existence of Riemann mappings presented in the previous section, we give an iterative construction of Riemann mapping on a bounded domain D. More precisely, given a bounded domain D⊆
Step 1. We imbed D into the unit disk Δ by complex affine mapping. There exists r>0 such that f1(z)=r(z-a) maps D into Δ with f1(z)=0 and f'1(a)>0. If D is a disc, then we get our results because f is clearly surjective for suitable r; if D is not a disc, we continue our construction.
Step 2. Choose a point w1∈Δ\f1(D) such that
$ {f_2}(z) = {\alpha _1}{\psi _{h\left( { - {w_1}} \right)}} \circ h \circ {\psi _{{w_1}}} \circ {f_1}(z), $ |
where
$ \left| {f_2^\prime (a)} \right| = \left| {f_1^\prime (a)} \right|\frac{{1 + \left| {{w_1}} \right|}}{{2\sqrt {\left| {{w_1}} \right|} }}. $ |
Note that f2 is not surjective since
Step 3. We construct fj+1 inductively, by repeating the method in Step 2. Assume that we have constructed fj. Choose wj∈Δ\fj(D) such that
$ {f_{j + 1}}(z) = {\alpha _j}{\psi _{h\left( { - {w_j}} \right)}} \circ h \circ {\psi _{{w_j}}} \circ {f_j}(z), $ |
where
$ \left| {f_{j + 1}^\prime (a)} \right| = \left| {f_j^\prime (a)} \right|\frac{{1 + \left| {{w_j}} \right|}}{{2\sqrt {\left| {{w_j}} \right|} }}. $ |
From the above steps we get a sequence of holomorphic injective mappings {fj:D→Δ} with
$ {f_j}(a) = 0,f_j^\prime (a) > 0. $ |
Theorem 2.1 Let {fj} be the sequence constructed above. Then {fj} converges to a biholomorhpic mapping g from D to the unit disk Δ such that g(a)=0 and g'(a)>0, uniformly on compact subsets of D.
Proof By the uniqueness of the Riemann mapping f:D→Δ with f(a)=0 and f'(a)>0, it suffices to show that any subsequence of {fj} has a subsequence that converges to the Riemann mapping f uniformly on compact sets of Δ.
By the Cauchy's inequality, there exists b>0 such that |f'j(a)|≤b∈
$ \left| {f_{j + 1}^\prime (a)} \right| = \left| {f_1^\prime (a)} \right|\prod\limits_{k = 1}^j {\frac{{1 + \left| {{w_k}} \right|}}{{2\sqrt {\left| {{w_k}} \right|} }}} , $ |
we get
$ \left| {{w_j}} \right| \to 1. $ |
Recall that we have
$ \left| {{w_j}} \right| < \frac{{1 + d\left( {0,\partial {f_j}(D)} \right)}}{2} \le 1 $ |
thus we can obtain that
$ d\left( {0,\partial {f_j}(D)} \right) \to 1. $ |
By Montel's theorem, any subsequence of {fj} has a subsequence, say {fjk}, that converges to a holomorphic function
Set gj=fj-1. By Montel's theorem again, we may assume {gjk} converges to a holomorphic function g on Δ uniformly on compact sets of Δ. Note that
$ \tilde f(z) = \mathop {\lim }\limits_{h \to + \infty } {f_{{j_k}}}(z) = \mathop {\lim }\limits_{h \to + \infty } {w_k} = w. $ |
Since w∈Δ is arbitrary,
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