## Nagoya Mathematical Journal

### 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.

#### Article information

Source
Nagoya Math. J., Volume 187 (2007), 1-33.

Dates
First available in Project Euclid: 4 September 2007

https://projecteuclid.org/euclid.nmj/1188913892

Mathematical Reviews number (MathSciNet)
MR2354553

Zentralblatt MATH identifier
1135.22020

#### Citation

Dorfmeister, Josef; Wu, Hongyou. Unitarization of loop group representations of fundamental groups. Nagoya Math. J. 187 (2007), 1--33. https://projecteuclid.org/euclid.nmj/1188913892

#### References

• I. Biswas, On the existence of unitary flat connections over the punctured sphere with given local monodromy around the punctures, Asian J. Math., 3 (1999), 333–344.
• J. Dorfmeister and G. Haak, On constant mean curvature surfaces with periodic metric, Pacific J. Math., 182 (1998), 229–287.
• J. Dorfmeister and G. Haak, Construction of non-simply connected CMC surfaces via dressing, J. Math. Soc. Japan, 55 (2003), 335–364.
• J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom., 6 (1998), 633–668.
• J. Dorfmeister and H. Wu, Construction of constant mean curvature $n$-noids from holomorphic potentials, Math. Z., to appear.
• I. C. Gohberg, A factorization problem in normed rings, functions of isometric and symmetric operators and singular integral equations, Russian Math. Surveys, 19 (1964), 63–114.
• W. Goldman, Topological components of spaces of representations, Invent. Math., 93 (1988), 557–607.
• Y. Katznelson, An Introduction to Harmonic Analysis, Dover, 1976.
• N. Schmitt, Constant mean curvature trinoids, preprint (2001).