Unitarization of loop group representations of fundamental groups

Abstract

In this paper, we give a characterization of the simultaneous unitarizability of any finite set of $\operatorname{SL}(2, \mathbb{C})$-valued functions on $\mathbb{S}^{1}$ and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an $\mathbb{S}^{1}$-family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.