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

Let Bn ⊂ ℝn and Sn ⊂ Rn+1 denote the Euclidean n-dimensional unit ball and sphere, respectively. The extrinsic k-energy functional is defined on the Sobolev space Wk,2 (Bn, Sn) as follows: Ekext(u) = ∫Bn |Δs u|2 dx when k = 2s, and Ekext(u) = ∫Bn|∇ Δs 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∗: Bn → Sn, defined by u∗(x) = (x/|x|,0), is a critical point of Ekext(u) provided that n ≥ 2k + 1. The main aim of this paper is to establish necessary and sufficient conditions on k and n under which u∗: Bn → Sn is minimizing or unstable for the extrinsic k-energy.