Carathéodory conjecture

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, [1]. Other early references are the Invited Lecture [3] of Stefan Cohn-Vossen at the International Congress of Mathematicians in Bologna and the book [2] by Wilhelm Blaschke. Carathéodory never committed the Conjecture into writing. In [1], John Edensor Littlewood mentions the Conjecture and Hamburger's contribution [10] as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in [5] the formal analogy of the Conjecture with the Four Vertex Theorem for plane curves. Modern references for the Conjecture are the problem list of Shing-Tung Yau in [6] and the book [7] of Marcel Berger, as well as the books [18] and [19], [20], [21].

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Mathematical content

The Conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space admits at least two umbilic points. In the sense of the Conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of umbilics. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.

Mathematical research on an approach by a local index estimate

For analytic surfaces, an affirmative answer to this conjecture was given in 1940 by Hans Ludwig Hamburger in a long paper published in three parts [10]. The approach of Hamburger was via a local index estimate for isolated umbilics, which he showed to imply the Conjecture in his earlier work [8], [9]. In 1943, a shorter proof was proposed by Gerrit Bol [11], see also [23], but, in 1959, Tilla Klotz found and corrected a gap in Bol's proof in [10]. Her proof, in turn, was announced to be incomplete in Hanspeter Scherbel's dissertation [13] (no results of that dissertation related to the Carathéodory conjecture were published for decades, at least nothing was published up to June 2009). Among other publications we refer to papers [14]—[16].

All the proofs mentioned above are based on a reduction of the Carathéodory conjecture to the following Loewner conjecture: the index of every isolated umbilic point is never greater than one. Roughly speaking, the main difficulty lies in resolution of singularities generated by umbilical points. All the above-mentioned authors resolve the singularities by induction on 'degree of degeneracy' of the umbilical point, but none of them was able to present the induction process clearly.

In 2002, Vladimir Ivanov revisited the work of Hamburger on analytic surfaces with the following stated intent [17]:

"First, considering analytic surfaces, we assert with full responsibility that Carathéodory was right. Second, we know how this can be proved rigorously. Third, we intend to exhibit here a proof which, in our opinion, will convince every reader who is really ready to undertake a long and tiring journey with us."

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).

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