A Moebius characterization of Veronese surfaces in \(S^n\) (Q5947414)

The authors work in the classical lightcone model of the conformal \(n\)-sphere. Using the canonical lift of a surface immersed without umbillics into the \(n\)-sphere, they derive a set of frame equations. They prove that the vanishing of the so called Möbius form for an immersed sphere without umbilics implies that, after a suitable Möbius transformation, the surface is minimal with constant Gaussian curvature with respect to the metrical geometry of the standard \(n\)-sphere. By results of Bryant, Calabi and Wallach it follows that the sphere is immersed as a Veronese surface.