Classification of branched Willmore spheres

TL;DR

This paper proves that Bryant's quartic is holomorphic for branched Willmore immersions, leading to a complete classification of branched Willmore spheres, especially spheres, by analyzing the asymptotic behavior of the conformal Gauss map.

Contribution

It establishes the holomorphicity of Bryant's quartic for branched Willmore immersions and fully classifies branched Willmore spheres, including the case of spheres.

Findings

Bryant's quartic is holomorphic for branched Willmore immersions.

The quartic vanishes for spheres, enabling classification.

Asymptotic expansion of the conformal Gauss map at branched points is a null straight line.

Abstract

Given a branched Willmore immersion from a closed Riemann surface, we show that Bryant's quartic is holomorphic. Consequently, this quartic vanishes when the underlying surface is a sphere and we obtain the full classification of branched Willmore spheres. To do so, we show that the asymptotic expansion in the $C^2$-topology of the conformal Gauss map at a branched point is a null straight line.