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

Abstract

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