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

We study sequences $f_k:\Sigma_k \to \R^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy $\mathcal{W}(f) \leq \Lambda$. 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 Möbius transformations $\sigma_k$, such that for a subsequence $\sigma_k \circ f_k \circ \phi_k \to f$ weakly in $W^{2,2}_{{\rm loc}}$ away from finitely many points. For $\Lambda \lt 8\pi$ the map $f$ is unbranched. If the $\Sigma_k$ diverge in moduli space, then we show $\liminf_{k \to \infty} \mathcal{W}(f_k) \geq \min(8\pi,\omega^n_p)$. Our work generalizes results in $\cite{K-S3}$ to arbitrary codimension.

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Published 8 June 2012