The symmetry of minimizing harmonic maps from a two dimensional domain to the sphere

Abstract

We show that minimizing harmonic maps from an annulus in $R^2$ to the sphere in $R^3$ that agree on the boundary with the map $u_0(x,y)=(x/r,y/r,0)$ – where $r^2=x^2+y^2$ – must be radially symmetric. This result combined with previous results of Jäger–Kaul [JK], Brezis–Coron [BC] and of Bethuel–Brezis–Coleman–Hélein [BBCH] shows that for any symmetrical domain in $R^2$ and any symmetrical boundary data with image lying in a closed hemisphere, minimizing harmonic maps must be radially symmetric. We also give an example showing that this no longer has to be true when the boundary data has its image lying in a neighborhood – however small it may be – of a closed hemisphere.

Résumé

On montre que les applications harmoniques minimisantes d’un domaine bidimensionnel vers la sphère sont nécessairement symétriques dès que leur trace est symétrique et à valeur dans un hémisphère fermé. On montre également que ceci devient faux lorsque la trace est à valeurs dans un voisinage arbitrairement petit d’un hémisphère.