Harmonic maps of finite uniton type into inner symmetric spaces

Abstract

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 ∩ K 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 twospheres (whose conformal Gauss maps take value in the non-compact symmetric space SO + (1, n + 3)/SO(1, 3) × 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.

A duality theorem for harmonic maps into non-compact symmetric spaces and compact symmetric spaces

“…The main goal of this section is to prove Theorem 1.1, which states that a harmonic map into a non-compact symmetric space induces naturally a harmonic map into the compact dual symmetric space. This result is of importance for the discussion of Willmore surfaces of finite uniton type [8], since it permits to apply the work of Burstall and Guest [3], originally only applicable to harmonic maps into compact symmetric spaces, to the investigation of Willmore surfaces in spheres. We will see that all Willmore 2-spheres are of finite uniton type (the monodromy matrices all are trivial and all quantities of geometric interest are Laurent polynomials).…”

“…To be concrete, using this duality theorem, we can classify the conformal Gauss maps of Willmore 2-spheres by classifying all compact dual harmonic maps of these harmonic maps [15]. This also shows how one can describe all harmonic maps of finite uniton type into non-compact symmetric spaces, the main topic of [8].…”

“…In [8] the notion of finite uniton type is investigated in much more detail. In particular, it is shown there that the definition above coincides with the definition of [3] and [14].…”

“…This theorem turns out to permit to apply certain well-known results for finite uniton harmonic maps into compact inner symmetric spaces [3] directly to harmonic maps into non-compact inner symmetric spaces, which basically is the content of the follow-up paper [8].…”

Construction of Willmore two-spheres via harmonic maps into $SO^+(1,n+3)/(SO^+(1,1)\times SO(n+2))$

“…The main idea of our work are based on the DPW method for harmonic maps [8,13] and the description of harmonic maps of finite uniton type [4,12,10]. The DPW method [8] gives a way to produce harmonic maps in terms some meromorphic 1-forms, i.e., normalized potentials.…”