Dressing preserving the fundamental group

In this note we consider the relationship between the dressing action and the holonomy representation in the context of constant mean curvature surfaces. We characterize dressing elements that preserve the topology of a surface and discuss dressing by simple factors as a means of adding bubbles to a class of non finite type cylinders.Comment: 36 pages, 1 figur

Harmonic maps of finite uniton type into non-compact inner symmetric spaces

Due to the efforts of many mathematicians, there has been a classification of harmonic two-spheres into compact (semi-simple) Lie groups as well as compact inner symmetric spaces. Such harmonic maps have been shown by Uhlenbeck, Burstall-Guest, Segal to have a finite uniton number. Moreover, the monodromy representation was shown to be trivial and to be polynomial in the loop parameter. We will introduce a general definition according to which such maps are called to be of finite uniton type. This paper aims to generalize results of [2] to harmonic maps of finite uniton type into a non-compact inner symmetric space. For this purpose, we first recall some basic results about harmonic maps of finite uniton type. Then we interpret the work of Burstall and Guest on harmonic maps of finite uniton type into compact (semi-simple) Lie groups in terms of the language of the DPW method. Moreover, to make the work of Burstall and Guest applicable to our setting we show that a harmonic map into a non-compact inner symmetric space $G/K$ shares the normalized potential as well as the meromorphic extended framing with a harmonic map into $U/(U\cap K^{\mathbb{C}})$, the compact dual of $G/K$. Thus we reduce the description of harmonic maps of finite uniton type into a non-compact inner symmetric space to the description of harmonic maps of finite uniton type into a compact inner symmetric space. Our main goal for the study of such harmonic maps is to provide a classification of Willmore two-spheres (whose conformal Gauss maps take value in the non-compact symmetric space $SO^+(1,n+3)/SO(1,3)\times SO(n)$). We will finish this paper by presenting the coarse classification of Willmore two-spheres in terms of their conformal Gauss maps [28] as well as examples of Willmore surfaces constructed by using [13] and the results of this paper.Comment: 33 pages. We divide the original paper into several papers due to the length and the confusions of topics. The present paper is to characterize all harmonic maps of finite uniton type into non-compact inner symmetric spaces by their normalized potential