In [3] John Edensor Littlewood mentions the conjecture and Hamburger's contribution[4] as an example of a mathematical claim that is easy to state but difficult to prove.
[15] The proof for smooth surfaces by Brendan Guilfoyle and Wilhelm Klingenberg, first announced in 2008, [16] was published in three parts [17] [18] [19] concluding in 2024, the centenary of the conjecture.
Their challenging proof involves techniques spanning a number of areas of mathematics, including neutral Kaehler geometry, higher codimension parabolic PDE and Sard-Smale theory.
The conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space needs to admit at least two umbilic points.
The invited address of Stefan Cohn-Vossen[20] to the International Congress of Mathematicians of 1928 in Bologna was on the subject and in the 1929 edition of Wilhelm Blaschke's third volume on Differential Geometry[21] he states: While this book goes into print, Mr. Cohn-Vossen has succeeded in proving that closed real-analytic surfaces do not have umbilic points of index > 2 (invited talk at the ICM in Bologna 1928).
[4] The approach of Hamburger was also via a local index estimate for isolated umbilics, which he had shown to imply the conjecture in his earlier work.
First he follows the way passed by Gerrit Bol and Tilla Klotz, but later he proposes his own way for singularity resolution where crucial role belongs to complex analysis (more precisely, to techniques involving analytic implicit functions, Weierstrass preparation theorem, Puiseux series, and circular root systems).
Hamburger’s umbilic index bound for analytic surfaces leads to restrictions on the position of the roots of certain types of holomorphic polynomials.
[29] The proof takes any holomorphic polynomial with the stipulated properties and constructs a real analytic surface with an isolated umbilic point.
The index is determined by the number of zeros of the polynomial that lie inside the unit circle, and then Hamburger’s bound yields the stated result.
[16] All of the geometric quantities referred to are defined with respect to the canonical neutral Kähler structure, for which surfaces can be both holomorphic and Lagrangian.
with a single umbilic point?” This is answered by Guilfoyle and Klingenberg:[18] the associated Riemann-Hilbert boundary value problem would be Fredholm regular.
By Fredholm regularity, for a generic convex surface close to a putative counter-example of the global Carathéodory Conjecture, the associated Riemann-Hilbert problem would have no solutions.
[33] The proof follows that of the global conjecture, but also uses more topological methods, in particular, replacing hyperbolic umbilic points by totally real cross-caps in the boundary of the associated Riemann-Hilbert problem.