Let M be a compact immersed surface in the unit sphere S3 with constant mean curvature H. Denote by φ the linear map from Tp(M) into Tp(M), φ = A - H/2 I, where A is the linear map associated to the second fundamental form and I is the identity map. Let φ denote the square of the length of φ. We prove that if ∥φ∥L(2) ≤ C, then M is either totally umbilical or an H(r)-torus, where C is a constant depending only on the mean curvature H.
- Mean curvature
- Totally umbilical