REGULARITY OF HARMONIC MAPS INTO POSITIVELY CURVED MANIFOLDS

Let M be a compact Riemannian manifold of dimension m, N a complete Amply connected δ-pinched Riemannian manifold of dimension n. There exists a constant d(n). It is proved that if m≤d(n), then every minimizing map from M into N is smooth in the interior of M. If m=d(n)+1, such a map has at most diserete singular set and in general the Hausdorff dimension of the singular set is at most m-d(n)-1.

Let M be a compact Riemannian manifold of dimension m, N a complete simply connected delta-pinched Riemannian manifold of dimension n. There exists a constant d(n). It is proved that if m less-than-or-equal-to d(n), then every minimizing map from M into N is smooth in the interior of M. If m = d(n) + 1, such a map has at most discrete singular set and in general the Hausdorff dimension of the singular set is at most m - d(n) - 1.