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Weierstrass type representation of Willmore surfaces in \(S^n\). (English) Zbl 1158.53336

Authors’ abstract: We reformulate the Euler-Lagrange equations of Willmore surfaces in \(S^n\) as the flatness of a family of certain loop algebra-valued 1-forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in \(S^n\) and minimal surfaces in constant curvature spaces \(S^n\), \({\mathbb R}^n\), \(H^n\), and prove that some special Willmore surfaces can be derived from minimal surfaces in \(S^n\), \({\mathbb R}^n\), \(H^n\).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E20 Harmonic maps, etc.
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References:

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