Willmore surfaces in the unit N-sphere

Let M² be a compact Willmore surface in the n-dimensional unit sphere. Denote by φ_ij ^α the tracefree part of the second fundamental form φ_ij ^α of M², and by ℍ the mean curvature vector of M². Let Φ be the square of the length of φ_ij ^α and H = |ℍ|. We prove that if 0 ≤ Φ ≤ C(1 + H²/8), where (C = 2 when n = 3 and C = 4/3 when n ≥ 4, then either Φ = 0 and M² is totally umbilic or Φ = (C(1 + H²/8). In the latter case, either n = 3 and M² is the Clifford torus or n = 4 and M² is the Veronese surface.