Let $\Sigma$ be a surface and $\Omega$ be an open subset of $\Sigma$. Suppose that $\Omega$ is homeomorphic to the open unit disk $\mathbb{D}$ and is relatively compact in $\Sigma$. I'm interested in the 'boundary behaviour' of maps $\varphi:\mathbb{D}\to\Omega$ when $\partial\Omega$ is locally connected.
When $\Sigma$ is the complex plane or the Riemann sphere, then the Caratheodory-Torhorst Theorem can be used to show that there exists a homeomorphism $\varphi:\mathbb{D}\to\Omega$ which can be continuously extended to $\overline{\varphi}:\overline{\mathbb{D}}\to\overline{\Omega}$, where the bar denotes closure in its respective space. (Here, the map $\overline{\varphi}$ need not be a homeomorphism.)
I want to know whether this is true on general surfaces.
If $\partial\Omega$, the boundary of $\Omega$, is locally connected, does there exist a homeomorphism $\varphi:\mathbb{D}\to\Omega$ which can be continuously extended to $\overline{\varphi}:\overline{\mathbb{D}}\to\overline{\Omega}$?
If this is not true in genral, when can we guarantee the existence of such $\varphi$ and $\overline{\varphi}$?
Best Answer
To expand my comments. The proof of the Caratheodory-Torhorst Theorem that I know is given in sections 15, 16 of Milnor's notes "Dynamics in one complex variable". The proof is quite long and I will not reproduce it here. Milnor proves extenstence of a continuous extension of Riemann mapping $\mathbb D\to \Omega\subset S^2$, where $\Omega$ is a simply-connected open subset with locally connected boundary in $S^2$. But if you look in his proof, it is local with one exception, in the proof of Theorem 15.3. There he uses two facts from classical complex analysis:
(Fatou Theorem): Suppose that $f: \mathbb D\to \mathbb C$ is a bounded holomorphic map. Then the radial limit $$ f(e^{i\theta}):=\lim_{t\to 1-} f(te^{i\theta}) $$ exists for almost all $\theta\in [0,2\pi]$.
(Theorem of F. and M. Riesz) In the setting of (1), for each $w\in \mathbb C$ the set $$ E=\{\theta\in [0,2\pi]: f(e^{i\theta})=w\} $$ has measure zero.
Using a trick with a ramified covering, Milnor proves that (1) and (2) still hold for conformal maps $f: \mathbb D\to \Omega$ for arbitrary open subsets $\Omega\subset S^2$.
Let us verify that (1) and (2) still hold for conformal maps $f: \mathbb D\to \Omega\subset \Sigma$, where $\Sigma$ is an arbitrary Riemann surface and $\Omega$ is a relatively compact open subset of $\Sigma$. It suffices to consider the case when $S$ is not homeomorphic to $S^2$ (since, due to the uniformization theorem) the matter reduces to the case of conformal maps to $S^2$ which is already done by Milnor.
Case 1: $\Sigma$ is noncompact. Then $\Sigma$ is a 1-dimensional complex Stein manifold (as proven by Behnke and Stein). hence, $\Sigma$ holomorphically embeds in ${\mathbb C}^N$ for some $N$. (Actually, $N=3$ suffices but we do not need this fact.) By composing with this embedding, our map $f$ becomes a bounded (since $\Omega$ is relatively compact) holomorphic map $$ F=(f_1,...,f_N): \mathbb D\to \mathbb C^N. $$ But (1) and (2) hold for each holomorphic function $f_k: \mathbb D\to \mathbb C$ (they are all bounded since $F$ is). Since the finite union of measure zero sets still has measure zero, (1) and (2) hold for the map $F$. Now, the rest of Milnor's proof goes through and we obtain the desired continuous extension of a conformal embedding $f: \mathbb D\to \Sigma$ (whose image is relatively compact and has locally connected boundary).
Case 2: $\Sigma$ is compact. The idea of the proof is similar to Milnor's. Since $\Sigma$ is not simply-connected, there exists a 2-hold holomorphic covering map $p: X\to \Sigma$, where $X$ is another Riemann surface. Since $\mathbb D$ is simply-connected, the map $f: \mathbb D\to \Omega\subset \Sigma$ lifts to a holomorphic embedding $\tilde f: \mathbb D\to X$. But $p^{-1}(\Omega)$ consists of two components, $\Omega_1, \Omega_2$. Assume that $\tilde f(\mathbb D)\subset \Omega_1$. Pick a point $w\in \Omega_2$ and set $Y:= X\setminus \{w\}$. This is a noncompact Riemann surface and $\tilde f: \mathbb D\to Y$ is a holomorphic embedding with relatively compact image (since $w$ does not belong to the closure of $\Omega_1$). Since $p$ is a covering map (of finite degree), the boundary of $\Omega_1$ is still locally connected. hence, by Case 1, $\tilde f$ extends to a continuous map $\tilde g: cl(\mathbb D)\to Y\subset X$. Composing $\tilde g$ with $p$ we obtain the desired continuous extension of the original map $f$. qed