Higher order Willmore hypersurfaces in Euclidean space

Abstract

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.