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We study sequences $f_k:\Sigma_k \to \R^n$ of conformally immersed, compactRiemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq\Lambda$. Assume that $\Sigma_k$ converges to $\Sigma$ in moduli space, i.e.$\phi_k^\ast(\Sigma_k) \to \Sigma$ as complex structures for diffeomorphisms$\phi_k$. Then we construct a branched conformal immersion $f:\Sigma \to \R^n$and M\"obius transformations $\sigma_k$, such that for a subsequence $\sigma_k\circ f_k \circ \phi_k \to f$ weakly in $W^{2,2}_{loc}$ away from finitely manypoints. For $\Lambda < 8\pi$ the map $f$ is unbranched. If the $\Sigma_k$diverge in moduli space, then we show $\liminf_{k \to \infty} {\cal W}(f_k)\geq \min(8\pi,\omega^n_p)$. Our work generalizes results in \cite{K-S3} toarbitrary codimension.

$W^{2,2}$-conformal immersions of a closed Riemann surface into $\R^n$