Willmore Surfaces of Constant Moebius Curvature

Details Willmore Surfaces of Constant Moebius Curvature Willmore Surfaces of Constant Moebius Curvature

2006-09-04

arxiv.org/abs/math/0609057v2

Annals of Global Analysis and Geometry 32(2007), No.3, 297-310

Mathematics

Differential Geometry

16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6) has been corrected

Scientific paper

10.1007/s10455-007-9065-9

We study Willmore surfaces of constant Moebius curvature $K$ in $S^4$. It is proved that such a surface in $S^3$ must be part of a minimal surface in $R^3$ or the Clifford torus. Another result in this paper is that an isotropic surface (hence also Willmore) in $S^4$ of constant $K$ could only be part of a complex curve in $C^2\cong R^4$ or the Veronese 2-sphere in $S^4$. It is conjectured that they are the only examples possible. The main ingredients of the proofs are over-determined systems and isoparametric functions.

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