Tag Archives: injective

The First Lemma in Riemann Mapping Theorem

\mathbf{Problem:} Suppose U is a simply connected domain in \mathbb{C}, U\neq \mathbb{C}. Then there exists a holomorphic function f:U\to \mathbb{D}, the unit disc, and f is injective.
\mathbf{Proof:} Suppose z_0\not\in U, then g(z)=z-z_0 never achieves zero on U. Since U is simply connected, there exists a holomorphic function h such that h^2(z)=g(z), \forall x\in U.
Choose a\in U, then h(a)=b\neq 0. From open map theorem, \exists\, r>0 such that B_r(b)\subset h(U). Then B_r(-b)\cap h(U)=\emptyset, otherwise \exists\, a_1\in B_r(b), a_2\in B_r(-b) such that h(a_1)=-h(a_2). Since g=h^2 is injective, one must have a_1=a_2, so B_r(b)\cap B_r(-b)\neq \emptyset, which means 0\in B_r(b)\cap B_r(-b)\in h(U). Contradiction.
So one can check

\displaystyle f(z)=\frac{r}{h(z)+b}

satisfies all the conditions.

\mathbf{Remark:}

  • Pay attention to the domain U=\mathbb{C}\backslash (-\infty, 0], which is simply connected. Some other proof may have flaws when dealing with this stuition.
  • Riemann Mapping Theorem was firstly stated in his PhD thesis. Riemann’s flawed proof depended on the Dirichlet principle (whose name was created by Riemann himself).  Karl Weierstrass found that this principle was not universally valid. Later, David Hilbert was able to prove that, to a large extent, the Dirichlet principle is valid under the hypothesis that Riemann was working with However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of U which are not valid for simply connected domains in general. See Riemann Mapping Theorem.
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