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Proceedings of the London Mathematical Society Advance Access originally published online on August 22, 2008
Proceedings of the London Mathematical Society 2009 98(2):427-444; doi:10.1112/plms/pdn034
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© 2008 London Mathematical Society

The Riemann mapping theorem for semianalytic domains and o-minimality

T. Kaiser

Department of Mathematics
University of Regensburg
Universitätsstr. 31
D-93040 Regensburg
Germany

Received 22 May 2007. Revision received 6 May 2008.

We consider the Riemann mapping theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (that is, biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result, we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-minimal structure, provided that at singular boundary points the angles of the boundary are irrational multiples of {pi}.

2000 Mathematics Subject Classification 03C64, 32B20, 30C20, 30E15, 30D60, 30D05, 37E35.

The research is supported by DFG-project KN202/5-2.


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