The pure braid group Γ of a quadruply-punctured Riemann sphere acts on the SL(2, ℂ)-moduli ℳ of the representation variety of such sphere. The points in ℳ are classified into Γ-orbits. We show that, in this case, the monodromy groups of many explicit solutions to the Riemann-Hilbert problem are subgroups of SU(2). Most of these solutions are examples of representations that have dense images in SU(2), but with finite Γ-orbits in ℳ. These examples relate to explicit immersions of constant mean curvature surfaces.
© de Gruyter 2009