Branch points of willmore surfaces

摘要

We consider Willmore surfaces in 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 [GRAPHICS] where theta(0) is an element of N is the maximal multiplicity. Examples of branched minimal surfaces show that this estimate is optimal up to the error epsilon > 0.