Regularity of minimizing harmonic maps into the sphere. (English) Zbl 0555.58011

Schoen, Richard; Uhlenbeck, Karen

Invent. Math. 78, 89-100 (1984).

Analysing their ”minimizing tangent map”-condition developed for the general regularity theory of energy minimizing harmonic maps, the authors introduce a number d(k), \(k\geq 2\), by \(d(2)=2\), \(d(3)=3\) and \(d(k)=entier (\min (k+1,6))\), \(k\geq 4\), and show that if \(n\leq d(k)\) then every minimizing map from an n-manifold M into \(S^ k\) is smooth in the interior of M. Moreover, if \(n=d(k)+1\), such a map has at most isolated singularities, and, in general, the singular set is closed and of Hausdorff dimension \(\leq n-d(k)-1\). As corollaries they obtain various Liouville-type theorems for minimizing maps \(R^ n\to S^ k\). In additon, the authors also derive a regularity theorem for minimizing maps into closed hemispheres.

Reviewer: G.Toth

MSC:

58E20	Harmonic maps, etc.
58E30	Variational principles in infinite-dimensional spaces

Keywords:

regularity theory; energy minimizing harmonic maps

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