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 into 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 local real analytic version of the conjecture has had a troubled history with published proofs 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 concluding in 2024. Their proof involves techniques spanning a number of areas of mathematics, including neutral Kähler geometry, parabolic PDEs, and Sard-Smale theory.
