The three obdurate conjectures of differential geometry

We explore the role of symmetry in three obdurate conjectures of differential geometry: the Carathéodory, 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.