Rigidity of minimal hypersurfaces of spheres with two principal curvatures

Let ν be a unit normal vector field along M . Notice that ν : M −→ S satisfies that 〈ν(m),m〉 = 0. For any tangent vector v ∈ TmM , m ∈ M , the shape operator A is given by A(v) = −∇̄vν, where ∇̄ denotes the Levi Civita connection in S. For every m ∈ M , A(m) defines a linear symmetric transformation from TmM to TmM ; the eigenvalues of this transformation are known as the principal curvatures of M at m ∈ M . We will denote by ∆ the laplacian on M ; ‖A‖2 will denote the square of the norm of the shape operator, notice that since the second fundamental form II is given by II(v) = 〈A(v), v〉, then ‖A‖2 = ‖II‖2.