A global rigidity theorem of surface with parallel mean curvature in sphere

By using the Sobolev inequalities of P.Li and L.p estimate, we set up a global rigidity theorem for compact surface M with parallel mean curvature vector and genus zero embedded in sphere S\+\{p+2\}(p≥1). Suppose that the Gauss curvature of M has a positive lower bound k. It is shown that there is a constant A depending only on H and k such that if ‖σ‖\+2A, where H is the mean curvature and σ is the square length of the second fundamental form of M, then M is a totally umbilical surface in the sphere S\+3(1).