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

TL;DR

This paper investigates the stability and minimality of the equator map as a critical point for higher order extrinsic energy functionals on spheres, establishing conditions based on dimension and order.

Contribution

It provides necessary and sufficient conditions for the equator map to be minimizing or unstable for higher order extrinsic energies, extending classical results to higher derivatives.

Findings

Equator map is critical for dimensions n ≥ 2k+1.

Conditions for stability depend on the relation between n and k.

Results generalize known stability criteria for classical energies.

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: $E_{k}^{{\rm ext}}(u)=\int_{B^n}|\Delta^s u|^2\,dx$ 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.