On the stability of the equator map for higher order energy functionals

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 \textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}\left(B^n,{\mathbb{S}}^n\right)$ as follows: $E_k^{\mathrm{ext}}(u)={\int }_{B^n}|{\Delta }^{s}u{|}^{2}dx$ when $k=2s$, and $E_k^{\mathrm{ext}}(u)={\int }_{B^n}|\nabla {\Delta }^{s}u{|}^{2}dx$ 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 $u^{\ast }:B^n\to {\mathbb{S}}^n$, defined by $u^{\ast }(x)=(x/|x|,0)$, is a critical point of $E_k^{\mathrm{ext}}(u)$ provided that $n\ge 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^{\ast }:B^n\to {\mathbb{S}}^n$ is minimizing or unstable for the extrinsic $k$-energy.

13 pages

References in corpus (1)

• Higher order energy functionals