Harmonic maps of finite uniton type into inner symmetric spaces

Abstract

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}:M\to G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be more concrete, we prove that every harmonic map of finite uniton type from any 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.

36 pages. Comments are welcome

References in corpus (1)

• Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Cited by in corpus (8)

• Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials
• Willmore surfaces in spheres via loop groups IV: on totally isotropic Willmore two-spheres in $S^6$
• Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms
• The dual superconformal surface
• Construction of Willmore two-spheres via harmonic maps into $S{O}^{+}(1,n+3)/(S{O}^{+}(1,1)\times SO(n+2))$
• Classification of Homogeneous Willmore Surfaces in $S^N$
• A duality theorem for harmonic maps into inner symmetric spaces
• Willmore surfaces in spheres: the DPW approach via the conformal Gauss map