Branched Willmore Spheres

TL;DR

This paper classifies branched Willmore spheres with limited branch points, linking them to minimal surfaces, and applies this to analyze singularities in the Willmore flow for certain spheres.

Contribution

It extends Bryant's classification of Willmore spheres and connects these to minimal surfaces with specific end multiplicities, providing new insights into Willmore flow singularities.

Findings

Classification of branched Willmore spheres with up to three branch points

Connection between Willmore spheres and minimal surfaces with bounded end multiplicities

Application to singularity analysis in Willmore flow for spheres with energy ≤ 16π

Abstract

In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in $\R ^ 3$ with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to the analysis of singularities of the Willmore flow of non-Willmore spheres with Willmore energy $ \mc {W} (f) \leq 16\pi$.