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Harmonic diffeomorphisms, minimizing harmonic maps and rotational symmetry. (English) Zbl 0686.58012
The aim of the paper is to investigate the harmonic diffeomorphisms of the unit ball ${B}^{n}$ of ${R}^{n}$ onto a Riemannian manifold. It is shown that for $n=2$ these diffeomorphisms are minimizing with respect to the energy functional over ${H}^{1}$ mappings with trace condition. For $n\ge 3$ it is given a sufficient condition for this property in terms of the metric tensor. The case of finitely many punctual singularities is also treated. Interesting applications, especially for $n=3$ and $n=4$, are indicated when the harmonic diffeomorphisms are SO(n)-equivariant in an appropriate sense. One answers some questions raised in papers of Gulliver, White, Jäger, Kaul, Baldes and Helein.
Reviewer: D.Motreanu

##### MSC:
 58E20 Harmonic maps between infinite-dimensional spaces 58E30 Variational principles on infinite-dimensional spaces 53C42 Immersions (differential geometry)