On the stability of the equator map for higher order energy functionals
Ali Fardoun, Stefano Montaldo, Andrea Ratto
TL;DR
This work analyzes the stability and minimizing properties of the equator map for extrinsic $k$-energy functionals $E_k^{\rm ext}$ on maps $B^n \to {\mathbb S}^n$. It develops a unified framework based on higher-order Hardy inequalities, introducing constants $\alpha_k(n)$ and $A_k(n)$ and a polynomial family $\mathcal{P}_k(n)$ that encodes stability thresholds. The main result states that the equator map $u^*$ is energy minimizing if $\mathcal{P}_k(n) \ge 0$ and unstable if $\mathcal{P}_k(n) < 0$, recovering the classical $k=1$ and $k=2$ cases and yielding concrete thresholds $n_k^*$ such that minimization holds for $n \ge n_k^*$ while instability occurs for $2k+1 \le n < n_k^*$. The appendix provides a computer-assisted method to determine $n_k^*$ and includes extensive data, highlighting the practical impact for higher-order variational problems and regularity theory of high-order harmonic maps.
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: $E_{k}^{\rm ext}(u)=\int_{B^n}|Δ^s u|^2\,dx$ when $k=2s$, and $E_{k}^{\rm ext}(u)=\int_{B^n}|\nabla Δ^s u|^2\,dx$ 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.