Let $B^n\subset { \mathbb R}^{ n}$ and ${ \mathbb S}^n\subset { \mathbb R}^{ n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The \textit{ extrinsic $k$-energy functional} is defined on the Sobolev space $W^{ k,2}\left (B^n,{ \mathbb S}^n \right )$ as follows: $E_{ k}^{ { \rm ext}}(u)=\int_{ B^n}|\Delta^s u|^2\,dx$ when $k=2s$, and $E_{ k}^{ { \rm ext}}(u)=\int_{ B^n}|\nabla \Delta^s u|^2\,dx$ when $k=2s+1$. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map $u^*: B^n \to { \mathbb S}^n$, defined by $u^*(x)=(x/|x|,0)$, is a critical point of $E_{ k}^{ { \rm ext}}(u)$ provided that $n \geq 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^*: B^n \to { \mathbb S}^n$ is minimizing or unstable for the extrinsic $k$-energy.
Categories: math.DG