Willmore surfaces in spheres via loop groups IV: on totally isotropic Willmore two-spheres in $S^6$

Abstract

Totally isotropic surfaces in $S^6$ are not necessarily Willmore surfaces. Therefore it is the first goal of this paper to derive a geometric characterization of totally isotropic Willmore two-spheres in $S^6$. This will naturally yield to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, we also obtain an algorithm to construct totally isotropic Willmore two-spheres in $S^6$. As an application, a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in $S^6$ is constructed, illustrating the theory of this paper.

arXiv comment26 pages, adding more explanations for the computations

References in corpus (5)

• Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods
• Willmore surfaces in spheres via loop groups $I$: generic cases and some examples
• Harmonic maps of finite uniton type into inner symmetric spaces
• Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials
• Classification of Willmore 2-spheres in the 5-dimensional sphere

Cited by in corpus (5)

• Willmore surfaces in spheres via loop groups $I$: generic cases and some examples
• Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials
• A duality theorem for harmonic maps into inner symmetric spaces
• Construction of Willmore two-spheres via harmonic maps into $S{O}^{+}(1,n+3)/(S{O}^{+}(1,1)\times SO(n+2))$
• Willmore surfaces in spheres: the DPW approach via the conformal Gauss map