Curvature estimates for complete and bounded submanifolds in a Riemannian manifold

Abstract:

Let $M$ be a complete $n$-dimensional submanifold in the $(2n - 1)$-dimensional Euclidean space, with scalar curvature bounded from below. Baikousis and Koufogiorgos proved that the sectional curvature of $M$ satisfies sup ${K_M} \geqslant {\lambda ^{ - 2}}$ if $M$ is contained in a ball of radius $\lambda$. We extend this result to the case that the ambient space is a complete simply connected Riemannian manifold of nonpositive curvature.