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

Abstract

Let $f:\mathbb{C}\rightarrow \mathbb{R}^3$ be complete Willmore immersion with $\int_Σ|A_f|^2<+\infty$. 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 $Σ_k$ is a sequence of closed Willmore surface embedded in $\mathbb{R}^3$ with $W(Σ_k)<C$, and if the conformal class of $Σ_k$ converges in the moduli space, then we can find a Möbius transformation $σ_k$, such that a subsequence of $σ_k(Σ_k)$ converges smoothly.