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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  extrinsic   $k$  -energy functional  is defined on the Sobolev space   $W^{k,2}\left (B^n,\mathbb{S} ^n \right )$   as follows:   $E_{k}^{{\textrm{ext}}}(u)=\int _{B^n}|\Delta ^s u|^2\ dx$   when   $k=2s$  , and   $E_{k}^{{\textrm{ext}}}(u)=\int _{B^n}|\nabla \Delta ^s u|^2\ dx$   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   $E_{k}^{{\textrm{ext}}}(u)$   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.

On the Stability of the Equator Map for Higher Order Energy Functionals