Willmore surfaces of constant Mobius curvature

We study Willmore surfaces of constant Mobius curvature Kappa in S-4. It is proved that such a surface in S-3 must be part of aminimal surface in R-3 or the Clifford torus. Another result in this paper is that an isotropic surface ( hence also Willmore) in S4 of constant K could only be part of a complex curve in C-2 congruent to R-4 or the Veronese 2- sphere in S-4. It is conjectured that they are the only possible examples. The main ingredients of the proofs are over-determined systems and isoparametric functions.MathematicsSCI(E)7ARTICLE3297-3103