Parallel mean curvature vector submanifolds in the hyperbolic space

Abstract

This thesis concerns with two applications of the Omori-Yau maximum principle for complete non-compact submanifolds whose Ricci curvature are bound from below. The first of these is a pinching theorem for complete parallel mean curvature submanifolds in the standard hyperbolic space while the second one is an extrinsic diameter theorem for bounded mean curvature submanifolds in the standard hyperbolic space. To obtain the pinching theorem for complete parallel mean curvature sub-manifolds in the standard hyperbolic space, we generalize the results due to Q.M. Cheng to certain class of submanifolds immersed isometrically in the standard hyperbolic space. In order to do this, we studied carefully the proof of Simons' inequality in the work of Chern, do Carmo and Kobayashi to obtain the generalized Simons' inequality mentioned in Santos' paper. By using this inequality together with the maximum principle of Yan-Omori, we obtained the pinching theorem for parallel mean curvature vector submanifolds in the standard hyperbolic space which parallels the results of Cheng. On the other hand, we studied the inequality on the Laplacian of the hyperbolic cosine of the distance function for a submanifold in the standard hyperbolic space. We discover that this inequality, when used together with the Omori type maximum principle, yields extrinsic diameter estimates for submanifolds in the standard hyperbolic space. As a corollary, one can recover the well-known result that there exists no compact constant mean curvature submanifolds in the standard hyperbolic space if |H| <= 1.