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The Scientific World Journal
Volume 2014, Article ID 697132, 8 pages
Research Article

Gap Phenomenon of an Abstract Willmore Type Functional of Hypersurface in Unit Sphere

College of Information System and Management, National University of Defense Technology, Changsha, Hunan 410073, China

Received 18 April 2014; Accepted 26 May 2014; Published 5 June 2014

Academic Editor: Luc Vrancken

Copyright © 2014 Yanqi Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called -Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the -Willmore hypersurface, for which the variational equation and Simons’ type integral equalities are obtained. Moreover, we construct a few examples of -Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.