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 . i (0 . i . n) be the ith mean curvature and Q n = . ni=0 (.1)i+1( n i ). n.i1 . 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.