Consider locally minimizing harmonic maps from D-dimensional Euclidean space into a compact Lie group G. When $D=3$ the general regularity theory due to Schoen-Uhlenbeck,

Schoen, Richard; Uhlenbeck, Karen A regularity theory for harmonic maps. J. Differential Geom. 17 (1982), no. 2, 307–335,

implies that at most isolated singularities are possible. When the target is $G=SU(N)$ and $N=2$, the target is $S^3$ and a more refined result applies,

Schoen, Richard; Uhlenbeck, Karen Regularity of minimizing harmonic maps into the sphere. Invent. Math. 78 (1984), no. 1, 89–100.

Thus, in the case D=3, locally minimizing harmonic maps into $G=SU(2)$ are everywhere regular (real analytic). In that paper the authors announced the same result for any $N\geq3$ in a forthcoming paper. At least to my knowledge this announced paper has never appeared. Is everywhere regularity in the case $D=3$ still true for any $N\geq3$? Is there a reference in the literature?


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