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 img401 be a closed set with two or more points. By the uniformization theorem there exists a Fuchsian group of Moebius transformations such that img403 is conformally equivalent to the quotient manifold img405. The universal covering map img407 is then given by img409, where img411 is the natural quotient map onto img405 and img415 is the conformal bijection between img403 and img405. We will show that there exists img421 such that
displaymath423
Considering img425, one obtains this from variational estimates.

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Clearly, the class of universal covering maps contains two extremal cases: The case where img403 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 img403 behaves like the corresponding Poincarè metric of img447. 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 img449 for which Theorem 1 holds.

18.10.2019, 02:10