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Regularity of weakly harmonic maps from a surface into a manifold with symmetries. (English) Zbl 0718.58019
Let N be a compact Riemannian homogeneous space. The author proves that every weakly harmonic map u from a Riemannian surface M into N is smooth (and so harmonic). That conclusion was known in the case $$N=S^ n$$ [the author, C. R. Acad. Sci., Paris, Sér. I 305, 565-568 (1987; Zbl 0641.49003)]; or under some additional hypotheses on u. J. Eells informed me that the author has recently obtained the above regularity result for any compact manifold N. That is a remarkable result. The proof here amounts to recognizing (by means of suitably constructed divergence free vector fields) that u is locally a solution of a certain Dirichlet problem on a 2-disk and so smooth. Perhaps some technicalities of the preliminary lemmas could be re-expressed more elegantly in terms of left invariant metrics and vector fields on homogeneous spaces.
Reviewer: A.Ratto (Coventry)

##### MSC:
 58E20 Harmonic maps, etc. 58J05 Elliptic equations on manifolds, general theory 35D10 Regularity of generalized solutions of PDE (MSC2000)
##### Keywords:
weakly harmonic map; regularity
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##### References:
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