Convex function on pseudo-Grassmann manifold and its applications for Bernstein-type theorem

We derive the Hessian estimate of the -function defined on the pseudo-Grassmann manifold , which is convex by the estimate. As an application, we give a direct proof of the Bernstein-type theorem due to Y. Xin [Manuscripta Math., 2000, 103: 191-202]. We also estimate the squared norm of the second fundamental form of a complete spacelike submanifold in pseudo-Euclidean space in terms of the -function and the mean curvature.