Explicit Connections with SU(2)-Monodromy

TL;DR

This paper investigates the monodromy groups of solutions to the Riemann-Hilbert problem for a four-punctured sphere, showing many are subgroups of SU(2) with dense images but finite orbits, relating to constant mean curvature surfaces.

Contribution

It provides explicit examples of monodromy groups in SU(2) with dense images and finite orbits, connecting representation theory to geometric surface immersions.

Findings

Many solutions have monodromy groups as subgroups of SU(2).

Some representations have dense images in SU(2) with finite orbits.

Examples relate to explicit immersions of constant mean curvature surfaces.

Abstract

The pure braid group \Gamma 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 \Gamma-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 \Gamma-orbits in M. These examples relate to explicit immersions of constant mean curvature surfaces.