The three obdurate conjectures of differential geometry

TL;DR

This paper investigates how symmetry influences three longstanding conjectures in differential geometry related to surfaces in 3D space-forms, revealing that relaxing symmetry assumptions invalidates these conjectures.

Contribution

It demonstrates that the conjectures fail under more general metrics when symmetry assumptions are removed, providing insight into the role of symmetry in these problems.

Findings

Symmetry is crucial for the validity of the three conjectures.

Relaxing symmetry assumptions leads to counterexamples.

The role of symmetry in the Carathéodory conjecture is particularly subtle.

Abstract

We explore the role of symmetry in three obdurate conjectures of differential geometry: the Carath\'eodory, the Willmore and the Lawson Conjectures. All three Conjectures concern surfaces in 3-dimensional space-forms, which have a high degree of symmetry. It is shown that this symmetry is broken and more general ambient metrics are considered, none of the Conjectures continue to hold. The subtle manner in which symmetry enters the first Conjecture is also explained in detail.