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

Abstract

Let $f:\mathbb{C}\to {\mathbb{R}}^3$ 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 Möbius transformation ${\sigma }_k$, such that a subsequence of ${\sigma }_k({\Sigma }_k)$ converges smoothly.

11pages

References in corpus (1)

• Weak limit of an immersed surface sequence with bounded Willmore functional