W2,2-conformal immersions of a closed Riemann surface into Rn

We study sequences f(k) : Sigma(k) -> R-n of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy W(f) <= Lambda. Assume that Sigma(k) converges to Sigma in moduli space, i.e., phi(k)* (Sigma(k)) -> Sigma as complex structures for diffeomorphisms phi(k). Then we construct a branched conformal immersion f : Sigma -> R-n and Mobius transformations sigma(k), such that for a subsequence sigma(k) circle f(k) circle sigma(k) -> f weakly in W-loc(2,2) away from finitely many points. For Lambda < 8 pi the map f is unbranched. If the Sigma(k) diverge in moduli space, then we show lim inf(k ->infinity) W(f(k)) >= min(8 pi, w(p)(n)). Our work generalizes results in [12] to arbitrary codimension.