On the Normality of Topological Target Manifolds for Riemann-Hilbert Problems

Abstract

In recent years R. Belch proposed an approach for investigating nonlinear Riemann-Hilbert problems with non-smooth target manifold. His main result is a characterization of solutions to Riemann-Hilbert problems as extremal functions in certain function classes. However, a complete analogy to corresponding results for problems with smooth target manifold holds only for a subclass of the toplogical target manifolds introduced by Belch, which are called normal. The conjecture that this subclass coincides with the whole class of topological target manifolds was left unproved. In the present paper we give a (counter-)example of a topological target manifold for which the solution set of the Riemann-Hilbert problem is in some sense bigger than in the smooth case. The problem to characterize normal topological target manifolds in geometric terms arises now as a challenging question of ongoing research.KeywordsRiemann-Hilbert problemconformal mappingMathematics Subject Classification (2000)30E25