Carathéodory conjecture
In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924. Carathéodory never committed the conjecture to writing, but did publish a paper on a related subject. In John Edensor Littlewood mentions the conjecture and Hamburger's contribution as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in the formal analogy of the conjecture with the four-vertex theorem for plane curves. Modern references to the conjecture are the problem list of Shing-Tung Yau, the books of Marcel Berger, as well as the books.
The conjecture has had a troubled history with published proofs in the analytic case which contained gaps. The proof for surfaces of Hölder smoothness by Brendan Guilfoyle and Wilhelm Klingenberg, first announced in 2008, was published in three parts by 2024, together with a summary of the proof. Their arguments involve techniques spanning a number of areas of mathematics, including neutral Kähler geometry, parabolic PDEs, and Sard-Smale theory. The role of symmetry in the Carathéodory conjecture, the Willmore conjecture and the Lawson conjecture has also been highlighted.
