Hypersurfaces with harmonic conformal gauss map in S⁴

Let x : Mⁿ → Sⁿ⁺¹ be an oriented hypersurface in Sⁿ⁺¹, the conformal Gauss map G = (H,Hx + e_n+1) : Mn → R₁ⁿ⁺³ is invariant under Möbius transformations of Sⁿ⁺¹, where H, e_n+1 are the mean curvature, the global unit normal vector field of x, respectively. In this paper, we study the oriented hypersurface x : M³ → S⁴ with harmonic conformal Gauss map, and we classify the hypersurfaces in S4 with constant Möbius scalar curvature under Möbius transformation group, which gives some examples of hypersurfaces with harmonic conformal Gauss map, but not Willmore hypersurfaces.