# Friday April 17th

Last time: extending a real-analytic function into $\CC$.

![](figures/image_2020-04-17-13-40-45.png)\

Theorem
:   Assume $\Omega$ is bounded and $\bd \Omega$ is a finite union of simple closed curves.
    Then the Riemann mappying function $f$ extends to a homeomorphism $\bar f: \bar \Omega \to \bar \DD$.


Proof:
Apply the previous theorem to the real analytic part of $\bd \Omega$, then map these diffeomorphically onto open arcs in $\bd \DD$.
Let $J_1, J_2$ be real analytic curves in $\bd \Omega$ meeting at $p$ and let $I_1, I_2$ denote their images in $\bd \DD$:

![](figures/image_2020-04-17-13-44-28.png)\

We want to show that $I_1$ meets $I_2$ and their endpoints $q_1, q_2$ coincide.
We have $\ell(r) = \int_{\gamma_r} \abs{f'}$, where $\gamma_r$ is as in the figure, and applying Cauchy-Schwarz yields
\begin{align*}
\abs{\ell(r)}^2 
&\leq_{CS} \int 1^2 \cdot \int \abs{f'}^2 \\
&\leq 2\pi r \int \abs{f'}^2 \\
\implies {\ell(r)^2 \over r} \leq 2\pi \int \abs{f'}^2
.\end{align*}

Then taking $\ell(r) \geq \delta$ for $\eps \leq r \leq R$  and integrating over $r\in (\eps, R)$ yields
\begin{align*}
\delta^2 \log\qty{R \over \eps} \leq 2\pi \iint_{\Omega(\eps, R)} \abs{f'}^2 = 2 \pi ~\mathrm{Area}(f(\Omega(\eps, R))) \leq 2\pi
\end{align*}
where $\Omega(\eps, R) = \Omega \intersect \theset{\eps \leq \abs{z-p} \leq R}$.

Since $\log\qty{R\over \eps} \converges{\eps\to 0}\to_? 0$ there exists a small $r>0$ such that $\ell(r) < \delta$.
Then $\abs{q_1 - q_2} < \delta$, so $q_1 = q_2$.

> Note that the classification of domains is nontrivial in higher dimension.

Next goal: Picard's theorem. 
Best proof is Picard's original, which comes from Ahlfors.

Recall the notion of covering maps from topology: for $\pi: E\to X$ is a covering map iff for every $p\in X$ there is a $U_x$ such that $\pi\inv(U) = \disjoint S_j \subset E$, where $\pi(S_j) = U$ is a homeomorphism.

Proposition
: If $E, X, Y$ are connected and locally path-connected and $E\mapsvia{\pi} X$ is a covering map, then if $Y$ is simply connected then any $f: Y\to X$ lifts to $\tilde f: Y\to E$.

Proposition
: Any continuous lift of a holomorphic map is also continuous.

Proof
:   Take $q\in U, p= f(q) \in \OO$.
    Let $V \ni p$ be a neighborhood such that $\pi\inv(V)$ satisfies the covering condition with the projections holomorphic homeomorphisms.
    Then $\tilde f(q) \in S_k$ for some $k$, and $\tilde f\inv(S_k) = f\inv V = U_1$ is a neighborhood of $q\in U$.
    So the restrictions of $\tilde f, \pi\inv \circ f$ to $U_q$ are equal and thus $\tilde f$ is holomorphic on each $U_q$.

Example: $e^z$ and $z^n$.

Proposition
: There exists a holomorphic covering map $\Phi: \DD \to \CC\setminus \theset{0, 1}$.

![](figures/image_2020-04-17-14-18-10.png)

Let $\omega = \exp({2\pi i \over 3})$, then define the LFT
\begin{align*}
\phi(z) = \omega{1 - z\over z - w^2}: \DD\to \HH
\end{align*}

which satisfies
\begin{align*}
\phi(1) = 0\\
\phi(\omega) = 1
\phi(\omega^2) = \infty
.\end{align*}

and the image of the above disc is

![](figures/image_2020-04-17-14-23-25.png)\

Then by the Riemann mapping theore, there is a holomorphic diffeomorphism $\psi: \Omega \to \DD$ and by the previous theorem, this lifts to $\bar \psi: \bar \Omega \to \bar \DD$, and $\psi$ can be chosen to fix $1, \omega$.
Conjugating by $\phi$ gives a holomorphic diffeomorphism $\Psi: \Omega^\sharp \to \HH$ which extends to a map $\bd \Omega^{\sharp}$ onto $\RR$ which fixes $0, 1$.

Idea: continue on a fractal-like manner to cover the unit disc.