Higher order Willmore hypersurfaces in Euclidean space

Let x: M ⁿ → R ⁿ⁺¹ be an n(≥ 2)-dimensional hypersurface immersed in Euclidean space R ⁿ⁺¹. Let σ _i (0 ≤ i ≤ n) be the ith mean curvature and Q _n = Σ _i=0 ⁿ (−1) ⁱ⁺¹( _i ⁿ )σ ₁ ⁿ⁻ⁱ σ _i . Recently, the author showed that W _n (x) = ∫ _M Q _n dM is a conformal invariant under conformal group of R ⁿ⁺¹ and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional W _n is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in R ⁴ which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper. The author is supported by Project No. 10561010 of NSFC