Computer-Assisted Proof of the Main Theorem of 'The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere'

TL;DR

This paper presents a computer-assisted proof confirming the holomorphy of specific meromorphic differentials in the classification of branched Willmore spheres in 3- and 4-spheres, utilizing Sage software.

Contribution

It introduces a novel computer-assisted approach to verify complex differential properties in geometric classification problems.

Findings

Confirmed holomorphy of quartic and octic differentials

Validated theoretical results using computational methods

Enhanced reliability of classification via computer verification

Abstract

We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the $3$-Sphere and the $4$-Sphere' (arXiv:1706.01405), using the free mathematical software Sage.