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

Let $f:\mathbb{C}\to {\mathbb{R}}^3$ be complete Willmore immersion with ${\int }_{\mathrm{\Sigma }}\mathrm{\mid }{A}_f{\mathrm{\mid }}^2<+\mathrm{\infty }$. 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 ${\mathrm{\Sigma }}_k$ is a sequence of closed Willmore surface embedded in ${\mathbb{R}}^3$ with $W({\mathrm{\Sigma }}_k)<C$, and if the conformal class of ${\mathrm{\Sigma }}_k$ converges in the moduli space, then we can find a M"obius transformation ${\sigma }_k$, such that a subsequence of ${\sigma }_k({\mathrm{\Sigma }}_k)$ converges smoothly.