Rigidity theorem for Willmore surfaces in a sphere

Let M2 be a compact Willmore surface in the (2 + p)-dimensional unit sphere S2 + p. Denote by H and S the mean curvature and the squared length of the second fundamental form of M2, respectively. Set ρ2 = S−2H2. In this note, we proved that there exists a universal positive constant C, such that if ∥ρ2∥2<C, then ρ2=0 and M2 is a totally umbilical sphere.