Author:  Leung, Yiuchung 
Title:  Parallel mean curvature vector submanifolds in the hyperbolic space 
Degree:  M.Phil. 
Year:  2001 
Subject:  Submanifolds Hyperbolic spaces Hong Kong Polytechnic University  Dissertations 
Department:  Dept. of Applied Mathematics 
Pages:  v, 56 leaves ; 30 cm 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b1573182 
URI:  http://theses.lib.polyu.edu.hk/handle/200/120 
Abstract:  This thesis concerns with two applications of the OmoriYau maximum principle for complete noncompact 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 submanifolds 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 YanOmori, 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 wellknown result that there exists no compact constant mean curvature submanifolds in the standard hyperbolic space if H <= 1. 
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