Full metadata record
| DC Field | Value | Language |
|---|---|---|
| dc.contributor | Department of Applied Mathematics | en_US |
| dc.creator | Leung, Yiu-chung | - |
| dc.identifier.uri | https://theses.lib.polyu.edu.hk/handle/200/120 | - |
| dc.language | English | en_US |
| dc.publisher | Hong Kong Polytechnic University | - |
| dc.rights | All rights reserved | en_US |
| dc.title | Parallel mean curvature vector submanifolds in the hyperbolic space | en_US |
| dcterms.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. | en_US |
| dcterms.extent | v, 56 leaves ; 30 cm | en_US |
| dcterms.isPartOf | PolyU Electronic Theses | en_US |
| dcterms.issued | 2001 | en_US |
| dcterms.educationalLevel | All Master | en_US |
| dcterms.educationalLevel | M.Phil. | en_US |
| dcterms.LCSH | Submanifolds | en_US |
| dcterms.LCSH | Hyperbolic spaces | en_US |
| dcterms.LCSH | Hong Kong Polytechnic University -- Dissertations | en_US |
| dcterms.accessRights | open access | en_US |
Files in This Item:
| File | Description | Size | Format | |
|---|---|---|---|---|
| b15731820.pdf | For All Users | 1.76 MB | Adobe PDF | View/Open |
Copyright Undertaking
As a bona fide Library user, I declare that:
- I will abide by the rules and legal ordinances governing copyright regarding the use of the Database.
- I will use the Database for the purpose of my research or private study only and not for circulation or further reproduction or any other purpose.
- I agree to indemnify and hold the University harmless from and against any loss, damage, cost, liability or expenses arising from copyright infringement or unauthorized usage.
By downloading any item(s) listed above, you acknowledge that you have read and understood the copyright undertaking as stated above, and agree to be bound by all of its terms.
Please use this identifier to cite or link to this item:
https://theses.lib.polyu.edu.hk/handle/200/120