Some remarks on Willmore surfaces embedded in $\mathbb{R}^3$

Abstract

Let $f:\mathbb{C}\rightarrow \mathbb{R}^3$ be complete Willmore immersion with $\int_{\Sigma}|A_f|^2<+\infty$. We will show that if $f$ is the limit of an embedded surface sequence, then $f$ is a plane. As an application, we prove that if $\Sigma_k$ is a sequence of closed Willmore surface embedded in $\mathbb{R}^3$ with $W(\Sigma_k)<C$, and if the conformal class of $\Sigma_k$ converges in the moduli space, then we can find a M\"obius transformation $\sigma_k$, such that a subsequence of $\sigma_k(\Sigma_k)$ converges smoothly.