Vol. 44, No. 2, 1973

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ISSN: 0030-8730
A constructive Riemann mapping theorem

Henry Cheng

Vol. 44 (1973), No. 2, 435–454

Classically, the Riemann mapping theorem states that any open, simply connected and proper subset of U of the complex plane is analytically equivalent to the open unit disk S(0,1). However this theorem is not constructively valid without some additional restriction on U. Two separate geometric conditions, mappability and maximal extensibility, on U are then proposed. The two conditions are shown to be mathematically equivalent. Finally the mappability condition is shown to be both necessary and sufficient for an analytic equivalence to exist constructively between U and S(0,1). The mappability condition is due Errett Bishop. The sufficiency proof is based on methods contained in [1].

Mathematical Subject Classification
Primary: 02E99
Secondary: 30A30
Received: 3 November 1971
Revised: 11 February 1972
Published: 1 February 1973
Henry Cheng