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. A proof for surfaces of Hölder smoothness , by Brendan Guilfoyle and Wilhelm Klingenberg, was first announced in 2008 , but was never published in its original form. Subsequently the authors published three other papers by 2024, which they claimed in a separate publication to establish the conjecture. Their arguments involve techniques from neutral Kähler geometry, parabolic PDEs, and Sard-Smale theory.
