an:06192430
Zbl 1271.53010
Kuwert, Ernst; Li, Yuxiang
\(W^{2,2}\)-conformal immersions of a closed Riemann surface into \(\mathbb{R}^n\)
EN
Commun. Anal. Geom. 20, No. 2, 313-340 (2012).
00306325
2012
j
53A30 53A07 49Q10
conformally immersed; Willmore energy; M??bius transformations
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 M??bius 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.