A boundary point whose complementary component contains another point is regular

Let $a\mathrm{\ne }\zeta$ lie in the same connected component $E$ of $\widehat{\mathbb{C}}\setminus \mathrm{\Omega }$ as $\zeta$, and let [given, construct] $T(w)=\frac{w-\zeta }{w-a}\mathrm{.}$ This Möbius map sends $\zeta$ to $0$, sends $a$ to $\mathrm{\infty }$, and maps $\mathrm{\Omega }$ biholomorphically onto the domain ${\mathrm{\Omega }}^{\mathrm{\prime }}=T(\mathrm{\Omega })$. Its image $E^{\mathrm{\prime }}=T(E)$ is a connected subset of $\widehat{\mathbb{C}}\setminus {\mathrm{\Omega }}^{\mathrm{\prime }}$ containing both $0$ and $\mathrm{\infty }$. Put $G=\widehat{\mathbb{C}}\setminus E^{\mathrm{\prime }}$. Then $G$ is an open connected neighbourhood of ${\mathrm{\Omega }}^{\mathrm{\prime }}$ and $0\mathrm{\notin }G$.

Let $\mathrm{\Gamma }$ be any cycle whose trace lies in $G$. By [L1], the index $n(\mathrm{\Gamma },\cdot )$ is locally constant on $\mathbb{C}\setminus {\mathrm{\Gamma }}^{\ast }$ and vanishes on the unbounded zero-index region. The connected set $E^{\mathrm{\prime }}\cap \mathbb{C}$ is disjoint from ${\mathrm{\Gamma }}^{\ast }$, and because $E^{\mathrm{\prime }}$ also contains $\mathrm{\infty }$, the local constancy from [L1] forces $n(\mathrm{\Gamma },p)=0$ for every $p\in E^{\mathrm{\prime }}\cap \mathbb{C}$. Since $E^{\mathrm{\prime }}=\widehat{\mathbb{C}}\setminus G$, this says exactly that $\mathrm{\Gamma }$ is null-homologous in $G$ by [L2]. Therefore $G$ is homologically simply connected.

Because $G$ is homologically simply connected and misses $0$, [L2] gives a holomorphic logarithm $L$ of the identity map on $G$, so $\exp (L(w))=w$ for $w\in G$. Hence [L2, step 1.1, construct] $\mathrm{Re}L(w)=\log \mathrm{\mid }w\mathrm{\mid }\mathrm{.}$ Because $T(z)\to 0$ as $z\to \zeta$ through $\mathrm{\Omega }$ and $0\mathrm{\notin }{\mathrm{\Omega }}^{\mathrm{\prime }}$, choose a small neighbourhood $V$ of $0$ with ${\mathrm{\Omega }}^{\mathrm{\prime }}\cap V\subseteq \{0<\mathrm{\mid }w\mathrm{\mid }<1\}$. On ${\mathrm{\Omega }}^{\mathrm{\prime }}\cap V$ the function $q(w)=\mathrm{Re}\frac{1}{L(w)}$ is harmonic, negative, and tends to $0$ as $w\to 0$ through ${\mathrm{\Omega }}^{\mathrm{\prime }}$, because $\mathrm{Re}L(w)=\log \mathrm{\mid }w\mathrm{\mid }\to -\mathrm{\infty }$. So $q$ is a weak local harmonic peak function at $0$ for the domain ${\mathrm{\Omega }}^{\mathrm{\prime }}$.

The composition $q\circ T$ is harmonic on $\mathrm{\Omega }\cap T^{-1}(V)$ by [L3], is negative there, and tends to $0$ as $z\to \zeta$ through $\mathrm{\Omega }$. Thus $q\circ T$ is a weak local subharmonic peak function at $\zeta$. Applying [L4] shows that $\zeta$ is regular for $\mathrm{\Omega }$.