The equator map and the negative exponential functional (Q1197351)

The study of stability properties of the equator map \(u^*: B^ n \to S^ n\), \(n \geq 3\), provided insight into the regularity theory of harmonic maps between Riemannian manifolds (here \(B^ n\) denotes the Euclidean unit \(n\)-ball, \(S^ n\) the Euclidean unit \(n\)-sphere and \(u^*(x)=({x\over | x|},0))\). In particular, \(u^*\) is a weakly harmonic map; i.e., a weak solution of the Euler-Lagrange equation associated with the energy functional \(E(u) = {1\over 2}\int| Du|^ 2\). The main aim of this paper is to replace \(E(u)\) by the negative- exponential energy functional \(NE(u)=\int e^{{1\over 2}| Du|^ 2}\); the equator map \(u^*\) is then shown to be an unstable critical point of \(NE(u)\) for all \(n \geq 2\).