The Riemann Mapping Theorem for semianalytic domains and o-minimality

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 (i.e. bihilomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class used by Ilyashenko in his work on Hilbert’s 16 problem if the angle of the domain at the boundary point to which 0 is mapped, is greater than 0. 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, supposed that given a singular boundary point, the angle of the boundary at this point is an irrational multiple of π. Introduction One of the central theorems in complex analysis is the Riemann Mapping Theorem Let Ω $ C be a simply connected domain in the plane. Then Ω can be mapped biholomorphically onto the unit ball B(0, 1). A nice overview of its proofs and their history can be found in Remmert [29]. ‘Riemann maps’ are in general transcendental and not algebraic functions. The goal of this paper is to show a ‘tame’ content of the Riemann Mapping Theorem. An important framework of ‘tame’ geometry is given by the category of semialgebraic sets and functions, semialgebraic sets and functions have very nice finiteness properties (see Bochnak et al. [3]). An outstanding and more general framework to study Mathematics Subject Classification (2000): 03C64, 32B20, 30C20, 30E15