Additional Problems for Chapter 13
-
Find explicit formulas for the hyperbolic metrics of the slit plane $U=\CC \sm (-\infty,-1]$ and the doubly-slit plane $V=\CC \sm ((-\infty,-1] \cup [1,+\infty))$. Show in particular that on the real locus of these domains the density of the hyperbolic metrics take the simple form
\[
\begin{aligned}
\rho_U (x) & = \frac{1}{2} \, \frac{1}{1+x} & (-\infty < x < -1) \\
\rho_V (x) & = \frac{1}{1-x^2} & (-1 < x < 1).
\end{aligned}
\]
-
Let $f$ be a rational function of degree $\geq 2$. Show that the group
\[
G_f = \{ \phi \in \Aut(\Chat) : \phi \circ f = f \circ \phi \}
\]
is finite. Can you describe $G_f$ when $f(z) = z^d$?
-
Every hyperbolic domain $U \subset \Chat$ has a conformal metric that is invariant under $\Aut(U)$. By contrast, verify that there is no conformal metric on $U=\Chat, \CC, \Cstar$ that is invariant under the action of the full group $\Aut(U)$.
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Verify that there is no closed hyperbolic geodesic in the punctured disk $\Dstar$. Compute the hyperbolic length of the non-trivial loop $\{ z: |z|=r \}$ in $\Dstar$ and find out how it behaves as $r \to 0^+$.
-
Show that the open set
\[
\Omega := \Big\{ z \in \HH: -\frac{1}{2} < \myre(z) < \frac{1}{2}, |z| > 1 \Big\}
\]
is a fundamental domain for the action of the modular group $\Gamma \cong \PSL_2(\ZZ)$ on $\HH$. This means
\[
\begin{aligned}
\HH & = \bigcup_{g \in \Gamma} g(\ov{\Omega}), \ \ \text{and} \\
g(\Omega) \cap \Omega & = \es \ \ \text{whenever} \ \ g \in \Gamma \sm \{ \text{id} \}.
\end{aligned}
\]
-
Show that the modular group $\Gamma$ is isomorphic to the free product $(\ZZ/2\ZZ) \ast (\ZZ/3\ZZ)$. In fact, it has the presentation $\langle A,B: B^2=(AB)^3=I \rangle$, where $A(z)=z+1$ and $B(z)=-1/z$.
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Suppose $u :\DD \to \RR$ is a harmonic function with $|u| \leq 1$. Show that for all $z \in \DD$,
\[
\| \nabla u (z) \| \leq \frac{4}{\pi} \ \frac{\cos(\pi u(z)/2)}{1-|z|^2} \leq \frac{4}{\pi} \ \frac{1}{1-|z|^2}.
\]
Compare Problem 25 of Chapter 7. (Hint: Apply the Schwarz lemma to a holomorphic function whose real part is $u$. You will need the formula for the hyperbolic metric of the vertical strip $\{ z: -1 < \myre(z) < 1 \}$.)
-
Prove the following general version of the Koebe distortion theorem: Suppose $U \subsetneq \Chat$ is simply connected and $f:U \to \CC$ is an injective holomorphic function. Let $K \subset U$ be compact with hyperbolic diameter $d$. Then
\[
\sup_{z,w \in K} \left| \frac{f'(z)}{f'(w)} \right| \leq e^{4d}.
\]
The classical distortion theorem corresponds to $U=\DD$, $K=[0,z]$, $d=\log((1+|z|)/(1-|z|))$.
-
Show that the modulus of the annulus $A= \CC \sm ((-\infty,-1] \cup [0,1])$ is $1/2$ by completing the following outline based on an idea of A. Douady:
(i) If $f: \{ z: 0 < \myre(z) < 1, 0 < \myim(z) < 1\} \to \HH$ is the unique conformal isomorphism which maps the boundary triple $(i,0,1)$ to $(-1,0,1)$, then $f(1+i)=\infty$. In fact, reflecting across the diagonal $\myre(z) = \myim(z)$ of the square is an anti-holomorphic involution that fixes $0,1+i$ and swaps $1,i$, so under $f$ it must correspond to the involution $z \mapsto -\ov{z}$ of $\HH$.
(ii) By the Schwarz reflection principle $f$ extends to an infinite degree covering map $f:\{ z: 0 < \myim(z) < 1 \} \to A$ whose deck group is generated by the translation $z \mapsto z+2$.
(iii) Since the deck group of the universal covering map $E(z)=e^{\pi i z}$ from $\{ z: 0 < \myim(z) < 1 \}$ to $A_{1/2} := \{ z: e^{-\pi} < |z| < 1 \}$ is also generated by $z \mapsto z+2$, it follows that $E(z) \mapsto f(z)$ is a well-defined conformal isomorphism between $A_{1/2}$ and $A$.
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Let $U \subsetneq \CC$ be a simply connected domain with the hyperbolic metric $\rho_U(z) |dz|$. For $z \in U$ and small $\ve>0$ let $m(z,\ve)$ denote the modulus of the annulus $U \sm \ov{\DD}(z,\ve)$. Prove that
\[
\rho_U(z) = \lim_{\ve \to 0} \ \frac{2}{\ve} \ e^{-2\pi \, m(z,\ve)}.
\]
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