Rigidity theorem for Willmore surfaces in a sphere

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