Partial Hölder continuity for Q-valued energy minimizing maps

We consider multivalued maps between $Ω\subset \mathbb{R}^N$ open ($N \ge 2$) and a smooth, compact Riemannian manifold $\mathcal{N}$ locally minimizing the Dirichlet energy. An interior partial Hölder regularity result in the spirit of R. Schoen and K. Uhlenbeck is presented. Consequently a minimizer is Hölder continuous outside a set of Hausdorff dimension at most $N-3$. F. Almgren's original theory includes a global interior Hölder continuity result if the minimizers are valued into some $\mathbb{R}^m$. It cannot hold in general if the target is changed into a Riemannian manifold, since it already fails for "classical" single valued harmonic maps.

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