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Willmore surfaces in spheres via loop groups. IV: On totally isotropic Willmore two-spheres in \(S^6\). (English) Zbl 1473.53086

Summary: In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in \(S^6\), which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in \(S^6\). This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in \(S^6\). This gives a negative answer to an open problem of N. Ejiri [Proc. Lond. Math. Soc. (3) 57, No. 2, 383–416 (1988; Zbl 0671.53043)]. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in \(S^6\).
For Part III, see [the author, Tohoku Math. J. (2) 69, No. 1, 141–160 (2017; Zbl 1368.53010)].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A31 Differential geometry of submanifolds of Möbius space
58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
53C35 Differential geometry of symmetric spaces
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References:

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