2007/05/31 by Tobias Kaiser
Mathematics · #Analytic and geometric function theory #Ball (mathematics) #Boundary (topology) #Bounded function #Complex plane #Discrete mathematics #Domain (mathematical analysis) #Geometric function theory #Geometry #Holomorphic and Operator Theory #Irrational number #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Riemann hypothesis #Riemann sphere #Riemann surface #math.LO #msc:03C64 #msc:30C20 #msc:30D05 #msc:30D60 #msc:30E15 #msc:32B20 #msc:37E35
paper · pdf · doi:10.1112/plms/pdn034
22 pages
arxiv created 2008/07/21 · openalex publication_date 2008/08/22 · arxiv updated 2014/02/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 π.