

Preprint 36/1998
Universal covering maps and radial variation
Peter W. Jones and Paul F. X. Müller
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Submission date: 02. Sep. 1998
Pages: 24
published in: Geometric and functional analysis, 9 (1999) 4, p. 675-698
DOI number (of the published article): 10.1007/s000390050099
Bibtex
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Abstract:
We let be a closed set with two or more points. By the uniformization theorem there exists a Fuchsian group of Moebius transformations such that
is conformally equivalent to the quotient manifold
. The universal covering map
is then given by
, where
is the natural quotient map onto
and
is the conformal bijection between
and
. We will show that there exists
such that
Considering , one obtains this from variational estimates.
Clearly, the class of universal covering maps contains two extremal cases: The case where 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
behaves like the corresponding Poincarè metric of
. 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
for which Theorem 1 holds.