So, I am studying complex analyisis in one variable from various books (Rudin, Forster and Gamelin) and each one of them uses a particular regularity condition to prove their theorems, and I am not sure if they are equivalent.
A point $p$ in the boundary of a Riemann Surface Y is regular if there is some continuous function $u : \bar{Y} \cap V \to \mathbb{R}$ where $V$ is a neighbourhood of $p$ such that $u$ is subharmonic in $Y \cap V$, $u(p)=0$ and $u(x)<0$ otherwise (Forster)
A point $p$ in the boundary of an open subset $U$ of $\mathbb{C}$ is simple if for every sequence $\{z_n\}$ in $U$ converging to $p$ there is some path $\gamma : [0,1] \to U$ and points $t_1<t_2< \ldots \to 1$ such that $\gamma (t_n)=z_n$ (Rudin)
An curve $\gamma$ contained in the boundary of a subset $U$ of $\mathbb{C}$ is a free analytic arc if for every point $p \in \gamma$ there is some conformal map from the unit disk to some neighbourhood $V$ of $p$ that maps the real line to $\gamma \cap V$ and the upper half plane to $U \cap V$ (Gamelin)
And they use these definitions to prove:
- The Dirichlet boundary problem can be solved on open subsets whose boundaries consist of regular points.
- Conformal maps from a simply connected, bounded subset $U$ of $\mathbb{C}$ to the unit disk whose boundary points are simple can be extended conformally to the $\bar{U}$.
- One can use Schwarz reflection principle and construct Green functions for domains with free analytic boundary (ie, each connected component of the boundary is a free analytic closed curve).
Is there any equivalence between the definitions? I already know that free analytic boundary implies regular boundary, and Caratheodory's theorem, ie, all the boundry points of a simply connected domain are simple if and only if the boundary is a Jordan curve, but cannot find more connections.
Of course, there is some ambiguity here because the definition of free analytic arcs depends on a neigbourhood of the boundary, so the question is not about the boundary of a Riemann surface alone, but about the boundary of a subset of a Riemann surface, because I would like to be able to construct an exhaustion of every Riemann surface using Runge compacts with the nicest properties on their boundary.
First of all, note that the subharmonic function in the definition of regular point in Forster is required to be negative, not positive, away from the point.
Every boundary point of a simply connected region $\Omega\subsetneq\mathbb{C}^*$ (the Riemann sphere) is regular (provided $\partial\Omega$ contains more than one point), even if its boundary points are not simple and $\partial \Omega$ does not contain any analytic arc. This is not hard, and is proven e.g. in Donald Marshall's Complex analysis, chapter 13 section 2. Similarly, there are simply connected regions on which every point is simple and yet the boundary does not contain any analytic arc: it suffices to take a jordan region bounded by an Osgood curve.
Thus, $(2)\not\rightarrow (1), (2)\not\rightarrow (3)$. Similarly, there are regions such that every boundary point is simple but not every point is regular: just take $\mathbb{C}-\{0\}$. The maximum principle implies that $0$ cannot be a regular point (in general, isolated points cannot be regular). So there is in general no relation between $(1)$ and $(2)$. On the other hand it is quite easy to see how $(1)$ and $(2)$ are implied by $(3)$, as you already noticed.