Weierstrass Type Representation of Willmore Surfaces in S ⁿ

Authors: Xia, Qiao¹; Shen, Yi²

Source: Acta Mathematica Sinica, Volume 20, Number 6, November 2004 , pp. 1029-1046(18)

Publisher: Springer

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Abstract:

In this paper, 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, R ⁿ, H ⁿ, and prove that some special Willmore surfaces can be derived from minimal surfaces in S ⁿ, R ⁿ, H ⁿ.

Keywords: Willmore surfaces; Extended lift of a conformal Willmore immersion; Loop group; Weierstrasstype representation; Minimal surface; 53C42; 58E20

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10114-004-0389-0

Affiliations: 1: Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China, Email: xiaqiaoling@zju.edu.cn 2: Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China, Email: yibingshen@zju.edu.cn

Publication date: 2004-11-01