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

Abstract

Abstract 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.