# Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit disc. In the following, let us call this map the "Riemann map" of $G$ (note that it is unique up to pre-composition by Möbius transformations.

The following is a famous theorem, usually attributed to Carathéodory. It has become particularly well-known in one-dimensional holomorphic dynamics, where it is the reason for interest in the local connectivity of sets such as the Mandelbrot set or Julia sets.

Continuous Extension Theorem. The Riemann map $\varphi$ extends continuously to the unit circle if and only if the boundary $\partial G$ is locally connected.

While Carathéodory's theory of the boundary behaviour of a conformal map, and the notion of prime ends (Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73 (1913), 323–370), is very relevant to this result, the result is not due to him. Indeed, the notion of local connectivity was introduced by Hans Hahn (and independently by Mazurkiewicz) only around the same time as Carathéodory's paper appeared. The theorem appears to have been stated first by Marie Torhorst in her Bonn doctoral thesis (supervised by Hahn) in 1918.

(EDIT. More precisely, she announces the following (along with other results) in the short introduction to the paper.

Theorem. The following are equivalent for any simply-connected plane domain $G$: "The boundary of $G$ is locally connected" and "all prime ends of $G$ are of the first kind".

The notion of prime ends had been introduced by Carathéodory explicitly to study the boundary behaviour of the Riemann map; in particular, his results show that $\varphi$ extends continuously if and only if every prime end of $G$ is "of the first kind". Hence, although Torhorst does not mention Riemann maps explicitly - she seems interested in topology, not complex analysis - the continuous extension theorem is equivalent to the stated theorem, and this would have been understood by anyone familiar with prime end theory.)

The thesis itself appears to have been lost, but her results (or, as she states, "some of the results of her thesis") appeared in 1921 (Über den Rand der einfach zusammenhängenden ebenen Gebiete, Math. Z. 9 (1921), no. 1-2, 44–65).

Torhorst carried out a careful study of the topological properties of prime ends, and the abovementioned theorem is a consequence of more detailed results, when combined with Carathéodory's paper. It appears that Torhorst's work was well-known in topology circles at least until about 1929, when Wilder, who was in Texas, wrote (in On a certain type of connected set which cuts the plane):

"C. Carathéodory has made an admirable analysis of the structure of the boundary of a simply connected domain. In all that follows, I shall assume familiarity with his work, as well as with the work of Miss Marie Torhorst in the same connection."

However, in later contributions to the theory of prime ends, her work appears to have been completely forgotten. As Sarason wrote in his unpublished paper from 1968, "On prime ends and local connectivity":

"It is regrettable that Torhorst's results have been overlooked by more recent workers in the theory of prime ends. Her paper is not mentioned in the book of Collingwood and Lohwater, nor in the paper of Ursell and Young, nor in the recent papers of Arsove, where some of her results reappear."

My question is:

Question. When did the Continuous Extension Theorem first enter the literature (after Torhorst's work), and when was it first attributed to Carathéodory?

Further background. The main result in Torhorst's paper states that each prime end impression of $G$ (or, equivalently, each cluster set of the map $\varphi$ at a point of the unit circle) contains at most two points where the boundary is locally connected, and at most three points where it is accessible from all sides in the sense of Schoenflies. Hence, if the boundary is locally connected at every point, then each cluster set is trivial and hence the map extends continuously. This proves the non-trivial part of the Continuous Extension Theorem.

I rediscovered the first part of the above statement as an undergraduate, and published an extension in Bull. London Math. Soc. in 2008 (unaware of Torhorst's work at the time). For further background, see the updated arXiv version of my article.

Interestingly enough, Torhorst's article is still sometimes cited, for another result, referred to as the Torhorst Theorem ("if $G$ is a complementary domain of a locally connected plane continuum $K$, then $\partial G$ is also locally connected."). This doesnot seem to be explicitly stated in her paper, although it follows easily from her results when combined with the previous work of Hahn and Schoenflies.

• the map you wrote is unique up to pre-composition (not post-composition) with a fractional linear map. – Alexandre Eremenko Sep 19 '14 at 12:55
• I don't know the answer to your question (sorry), but I'm curious : has there been any progress done towards proving MLC by studying the possible extension of the Riemann map of the complement of M? – Malik Younsi Sep 19 '14 at 15:48
• @MalikYounsi - It is rather the case that the continuous extension of the Riemann map is the real reason we are interested in MLC (as it would give us a complete topological model of the Mandelbrot set, as well as combinatorial rigidity and density of hyperbolicity). – Lasse Rempe-Gillen Sep 19 '14 at 16:10
• Thanks for the interest. Posting from mobile so apologies for typos. 1. Carathéodory considered SIMPLE closed curves, not general curves. 2. Carathéodory proves continuous extension iff all prime ends of first kind. Torhorst proves and states all prime ends of first kind iff lc. – Lasse Rempe-Gillen May 23 '15 at 7:33
• Not sure if Torhorst explicitly mentions continuous extension but this is clearly understood. Will try to clarify in question when I get the chance. – Lasse Rempe-Gillen May 23 '15 at 7:35