Simons-type inequalities for the compact submanifolds in the space of constant curvature

Abstract

For the compact submanifold M immersed in the standard Euclidean sphere S^n+p or the Euclidean space R^n+p, we obtain Simons-type inequalities about the first eigenvalue λ₁ and the squared norm of the second fundamental form S respectively. In particular, for the case of the ambient space is S^n+p, we need not the assumption that M is minimal. Following which, we obtain the estimate about the lower bound for S if it is constant respectively.