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MiS Preprint

36/1998

Universal covering maps and radial variation

Peter W. Jones and Paul F. X. Müller

Abstract

We let $E \subseteq \mathbb{C}$ be a closed set with two or more points. By the uniformization theorem there exists a Fuchsian group of Moebius transformations such that $\mathbb{C} \backslash E$ is conformally equivalent to the quotient manifold $\mathbb{D} / G$. The universal covering map $P: \mathbb{D} \to \mathbb{C} \backslash E$ is then given by $P=\tau o \pi$, where $\pi$ is the natural quotient map onto $\mathbb{D}/G$ and $\tau$ is the conformal bijection between $\mathbb{C} \backslash E$ and $\mathbb{D}/G$. We will show that there exists $e^{i \beta} \in \mathbb{T}$ such that $\int_0^1 |P^\pi (re^{i\beta})|dr<\infty$. Considering $u=log|P`|$, one obtains this from variational estimates. Theorem 1 There exists $e^{i\beta} \in \mathbb{T}$ and $M>0$ such that for $r<1$ $u(re^{i\beta})<-\frac{1}{M} \int_0^r |\nabla u (\rho e^{i\beta})|d\rho + M$. Clearly, the class of universal covering maps contains two extremal cases: The case where $\mathbb{C} \backslash E$ is simply connected and the case where E consists of two points. (We considered the simply connected case in an earlier paper where we solved Anderson\'s conjecture. The second case follows from well known estimates for the Poincarè metric on the triply punctured sphere.) In the course of the proof of Theorem 1 we measure the thicknes of E at all scales, and we are guided by the following philosophy. If, at some scale, the boundary E appears to be thick then, locally, the universal covering map behaves like a Riemann map. On the other hand, if E appears to be thin, then, locally, the Poincarè metric of $\mathbb{C} \backslash E$ behaves like the corresponding Poincarè metric of $\mathbb{C} \backslash \{0,1\}$. With the right estimates for the transition from the thick case to the thin case, this philosophy leads to a rigorous proof. Our proof also shows the existence of a very large set of angles $\beta$ for which Theorem 1 holds.