Willmore surfaces in the unit N-sphere

Abstract

Let M-2 be a compact Willmore surface in the n-dimensional unit sphere. Denote by phi(ij)(alpha) the tracefree part of the second fundamental form h(ij)(alpha) of M-2, and by H the mean curvature vector of M-2. Let Phi be the square of the length of phi(ij)(alpha) and H = H. We prove that if 0 < Phi < C(1 + (H2)/(8)), where C = 2 when n = 3 and C = (4)/(3) when n greater than or equal to 4, then either Phi = 0 and M-2 is totally umbilic or Phi = C(1 + (H2)/(8)) In the latter case, either n = 3 and M-2 is the Clifford torus or n = 4 and M-2 is the Veronese surface.