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

TL;DR

This paper investigates the behavior of Willmore surfaces in three-dimensional space, showing that certain limits are planes and establishing conditions under which sequences of embedded Willmore surfaces converge smoothly after Möbius transformations.

Contribution

It proves that complete Willmore immersions with finite total curvature limit to planes if they are limits of embedded surfaces, and establishes convergence criteria for sequences of embedded Willmore surfaces with bounded Willmore energy.

Findings

Limit of embedded surfaces is a plane.

Sequences with bounded Willmore energy and converging conformal class have smooth subsequential limits.

Provides conditions for smooth convergence of embedded Willmore surfaces.

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.