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

We study sequences $f_k:{\mathrm{\Sigma }}_k\to {\mathbb{R}}^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq \Lambda$\mathrm{.}Assumethat$\Sigma_k$convergesto$\Sigma$inmodulispace,i\mathrm{.}e\mathrm{.}$\phi_k^\ast(\Sigma_k) \to \Sigma$ascomplexstructuresfordiffeomorphisms$\phi_k$\mathrm{.}Thenweconstructabranchedconformalimmersion$f:\Sigma \to \R^n$andM\ddot{o}biustransformations$\sigma_k$,suchthatforasubsequence$\sigma_k \circ f_k \circ \phi_k \to f$weaklyin$W^{2,2}{loc}$ away from finitely many points. For $\mathrm{\Lambda }<8\pi$ the map $f$ is unbranched. If the ${\mathrm{\Sigma }}_k$ 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} to arbitrary codimension.