The Riemann Mapping Theorem for semianalytic domains and o-minimality

TL;DR

This paper demonstrates that the Riemann map for certain semianalytic domains can be realized within a quasianalytic class and is definable in an o-minimal structure under specific boundary angle conditions.

Contribution

It establishes the definability of the Riemann map in an o-minimal structure for semianalytic domains with irrational boundary angles, extending previous work on quasianalytic classes.

Findings

Riemann map germ can be realized in a quasianalytic class for boundary angles > 0.

The Riemann map is o-minimal definable for domains with irrational boundary angles.

The result connects quasianalytic classes with o-minimal structures in complex analysis.

Abstract

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. 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 $\pi$.