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\(W^{2,2}\)-conformal immersions of a closed Riemann surface into \(\mathbb{R}^n\). (English) Zbl 1271.53010
Summary: We study sequences \(f_k:\Sigma_k\to\mathbb R^n\) 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 \(f:\Sigma\to\mathbb R^n\) and Möbius transformations \(\sigma_k\), such that, for a subsequence \(\sigma_k\circ f_k\circ\phi_k\to f\) weakly in \(W^{2,2}_{\mathrm{loc}}\) 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 \(\lim\inf_{k\to\infty}\mathcal W(f_k)\geq\min(8\pi,\omega^n_p)\). Our work generalizes results in one of our recent preprints to arbitrary codimension.

53A30 Conformal differential geometry (MSC2010)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
49Q10 Optimization of shapes other than minimal surfaces
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