Branch points of Willmore surfaces

Abstract

We consider Willmore surfaces in ${\mathbb{R}}^3$ with an isolated singularity of finite density at the origin. We show that locally, the surface is a union of finitely many multivalued graphs, each with a unique tangent plane at zero and with second fundamental form satisfying ${\displaystyle |A(x)|\le C_{\epsilon }|x{|}^{-1+1/{\theta }_0-\epsilon },\forall \epsilon >0,}$ where ${\theta }_0\in \mathbb{N}$ is the maximal multiplicity. Examples of branched minimal surfaces show that this estimate is optimal up to the error $\epsilon >0$