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 (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.

• Publication year

  2007

• Venue

  Proceedings of the London Mathematical Society

• Publication date

  2007-05-14

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1112/plms/pdn034arXiv 0705.1851
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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