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

We study sequences $f_k:Σ_k \to \R^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq Λ$. Assume that $Σ_k$ converges to $Σ$ in moduli space, i.e. $ϕ_k^\ast(Σ_k) \to Σ$ as complex structures for diffeomorphisms $ϕ_k$. Then we construct a branched conformal immersion $f:Σ\to \R^n$ and Möbius transformations $σ_k$, such that for a subsequence $σ_k \circ f_k \circ ϕ_k \to f$ weakly in $W^{2,2}_{loc}$ away from finitely many points. For $Λ< 8π$ the map $f$ is unbranched. If the $Σ_k$ diverge in moduli space, then we show $\liminf_{k \to \infty} {\cal W}(f_k) \geq \min(8π,ω^n_p)$. Our work generalizes results in \cite{K-S3} to arbitrary codimension.

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