Willmore Surfaces of Constant Moebius Curvature
Abstract
Description
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.
16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6) has been corrected
16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6) has been corrected