Harmonic maps of finite uniton number into inner symmetric spaces via based normalized extended frames

TL;DR

This paper develops a loop group framework for harmonic maps of finite uniton number into inner symmetric spaces, extending existing theories to non-compact targets and providing a method to construct all such maps.

Contribution

It transforms the Burstall-Guest theory into the DPW framework, enabling explicit construction of harmonic maps into both compact and non-compact symmetric spaces.

Findings

Establishes a 1-1 correspondence between finite uniton harmonic maps and normalized potentials.

Shows all such harmonic maps have normalized potentials in a nilpotent Lie algebra.

Provides a method to construct all finite uniton harmonic maps explicitly.

Abstract

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ of finite uniton number, from a Riemann surface $M,$ compact or non-compact, into inner symmetric spaces of compact or non-compact type. As a main result we show that the theory of [Burstall-Guest, Math Ann, 97], largely based on Bruhat cells, can be transformed into the DPW theory which is mainly based on Birkhoff cells. Moreover, it turns out that the potentials constructed in [Burstall-Guest, Math Ann, 97], mainly see section 5, can be used to carry out the DPW procedure which uses essentially the fixed initial condition $e$ at a fixed base point $z_0$. This extends work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces as target spaces (as a consequence of a "Duality Theorem"). It also permits to say that there is a 1-1-relation between finite uniton number harmonic maps and normalized potentials of a very specific and very controllable type. In particular, we prove that every harmonic map of finite uniton type from any (compact or non-compact) Riemann surface into any {compact or non-compact} inner symmetric space has a normalized potential taking values in some nilpotent Lie subalgebra, as well as a normalized frame with initial condition identity. This provides a straightforward way to construct all such harmonic maps. We also illustrate the above results exclusively by Willmore surfaces, since this problem is motivated by the study of Willmore two--spheres in spheres.