  We consider critical points of the geometric obstacle problem on vectorial
maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[
  \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N
\backslash \mathbb{B}^N(0)$}. \] Our main result is $C^{1,\alpha}$-regularity
for any $\alpha < 1$.
  Technically, we split the map $u=\lambda v$, where $v: \mathbb{B}^2 \to
\mathbb{S}^{N-1}$ is the vectorial component and $\lambda = |u|$ the scalar
component measuring the distance to the origin. While $v$ satisfies a weighted
harmonic map equation with weight $\lambda^2$, $\lambda$ solves the obstacle
problem for \[
  \int_{\mathbb{B}^2} |\nabla \lambda|^2+\lambda^2 |\nabla v|^2, \quad
\mbox{subject to $\lambda \geq 1$}. \] where $|\nabla v|^2 \in
L^1(\mathbb{B}^2)$. We then play ping-pong between the increases in the
regularity of $\lambda$ and $v$ to obtain finally the $C^{1,\alpha}$-result.
