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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.

##### MSC:
 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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