On closed minimal submanifolds in pinched Riemannian manifolds

Abstract

In this paper, we first prove a generalized Simons integral inequality for closed minimal submanifolds in a Riemannian manifold. Second, we prove a pinching theorem for closed minimal submanifolds in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by S. S. Chern, M. do Carmo, and S. Kobayashi and A. M. Li and J. M. Li respectively. Finally, we obtain a distribution theorem for the square norm of the second fundamental form of M M under the assumption that M M is a minimal submanifold with parallel second fundamental form in a Riemannian manifold.