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

Abstract

We study sequences $f_k:{\Sigma }_k\to {\mathbb{R}}^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy $\mathcal{W}(f)\le \Lambda$. Assume that ${\Sigma }_k$ converges to $\Sigma$ in moduli space, i.e. ${\varphi }_k^{\ast }({\Sigma }_k)\to \Sigma$ as complex structures for diffeomorphisms ${\varphi }_k$. Then we construct a branched conformal immersion $f:\Sigma \to {\mathbb{R}}^n$ and Möbius transformations ${\sigma }_k$, such that for a subsequence ${\sigma }_k\circ f_k\circ {\varphi }_k\to f$ 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 ${\mathrm{liminf}}_{k\to \infty }\mathcal{W}(f_k)\ge \min (8\pi ,{\omega }_p^n)$. Our work generalizes results in \cite{K-S3} to arbitrary codimension.

arXiv comment19 pages