Gap phenomenon of an abstract Willmore type functional of hypersurface in unit sphere.

Guohua Wu,Jin Liu,Yanqi Zhu

doi:10.1155/2014/697132

Guohua Wu, Jin Liu + Show 1 more

Open Access

https://doi.org/10.1155/2014/697132

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Abstract

For an n-dimensional hypersurface 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 hypersurface, for which the variational equation and Simons' type integral equalities are obtained. Moreover, we construct a few examples of W (n,F)-Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.

Highlights

Let φ : Mn → Sn+1(1) be an n-dimensional compact without boundary hypersurface in unit sphere
For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called W(n,F)-Willmore functional, which generalizes the well-known classic Willmore functional
Choose an orthonormal frames field {e1, . . . , en, en+1} along M such that {e1, . . . , en} are tangent to M and {en+1} is normal to M. Their dual frames are {θ1, . . . , θn} and {θn+1}, respectively; obviously, θn+1 = 0 when it is restricted over M

Summary

Introduction

Let φ : Mn → Sn+1(1) be an n-dimensional compact without boundary hypersurface in unit sphere. In [13], Guo and Li considered the W(n,1)-Willmore functional for submanifolds in unit sphere Sn+1(1): W(n,1) (φ) = ∫ ρdV They calculated the first variation and obtained Simons’ type inequalities and classified the point-wise gap phenomenon. It is well known that Simons’ integral inequality plays an important role in the study of minimal hypersurface It says that if M is an n-dimensional compact minimal hypersurface in (n + 1)-dimensional unit sphere Sn+1(1), . For classic Willmore hypersurface, Li [6] proved a Simons’ type integral result which says if M is an ndimensional compact classical Willmore hypersurface in unit sphere Sn+1(1), . Let M be an n-dimensional compact W(n,F)Willmore hypersurface in sphere Sn+1(1); we have Simons’ type equality and can give a discussion according to the sign of F, F󸀠, and F󸀠󸀠:.

Preliminaries

Variation Calculation and Examples

Simons’ Type Equalities and Gap Phenomenon