Real Algebraic and Analytic Geometry
Submission: 2008, January 3.
The proof of the Riemann mapping theorem is not constructive. We study versions of it for sets and functions which are definable in an $o$-minimal expansion of the real field. The diffeomorphisms between the subsets and the unit-ball can be chosen definable if we only request them to be continuously differentiable. For many structures expanding the real exponential field we can choose them smooth. For the globally subanalytic structure the diffeomorphisms can be chosen analytic and definable in an $o$-minimal expansion of it.
Mathematics Subject Classification (2000): 14P10,14P15,03C64.
Keywords and Phrases: Riemann mapping theorem, $o$-minimal structures.
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