Variational problem of an abstract Willmore type functional of sub-manifold in unit sphere

Abstract

For an n -dimensional sub-manifold in unit sphere, we introduce an abstract Willmore type called W ( n,F ) -Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the W ( n,F ) -Willmore sub-manifold, for which the variational equation and the Simons' type integral inequalities are obtained. Moreover, for some particular functions F , we construct a few examples of W ( n,F ) - Willmore sub-manifold. What's more important, a gap phenomenon characterization is given by use of Simons' type integral formula.