Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials

Abstract

Applying the DPW version of the theory developed by Burstall and Guest for harmonic maps of finite uniton type, we derive a coarse classification of Willmore two-spheres in $S^{n+2}$ in terms of the normalized potential of their (harmonic) conformal Gauss maps. Moreover, for the case of $S^6$, some geometric properties of the corresponding Willmore two-spheres are discussed. The classical classification of Willmore two-spheres in $S^4$ are also derived as a corollary.

20 pages, we add more discussions on loop group theory and simplify the proof of main theorem according to the referee's suggestions. arXiv admin note: text overlap with arXiv:1305.2514

References in corpus (3)

• Harmonic maps of finite uniton type into inner symmetric spaces
• 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

Cited by in corpus (4)

• 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
• A duality theorem for harmonic maps into inner symmetric spaces
• Construction of Willmore two-spheres via harmonic maps into $S{O}^{+}(1,n+3)/(S{O}^{+}(1,1)\times SO(n+2))$