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

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}:M\to G\mathrm{/}K$ 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.