Constantin Carathéodory (1873–1950) conjectured that, on every sphere-homeomorphic surface in the Euclidean 3-space, there necessarily exist at least two umbilic points. In 1940, an affirmative answer to this conjecture was given by Hans Ludwig Hamburger in [Ann. Math. (2) 41, 63–86 (1940; Zbl 0023.06902); Acta Math. 73, 175–228 (1941; Zbl 0024.17601) and Acta Math. 73, 229–332 (1941; Zbl 0025.42404)]. Both the conjecture and the proof were mentioned in the well-known book [John Edensor Littlewood, A mathematician’s miscellany, Methuen & Co, London (1953; Zbl 0051.00101)] as an example of an easy to state theorem with extremely long proof. In 1943, a shorter proof was proposed by Gerrit Bol [Math. Z. 49, 389–410 (1944; Zbl 0028.42501)], but, in 1959, Tilla Klotz found and corrected a gap in Bol’s proof [Commun. Pure Appl. Math. 12, 277–311 (1959; Zbl 0091.34301)]. Her proof, in turn, was announced to be incomplete in [H. Scherbel, A new proof of Hamburger’s index theorem on umbilical points, Dissertation no. 10281, ETH, Zürich].
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.
The aim of the paper under review is to give a new incite into the problem and to provide the reader with new arguments showing that the inductive process of singularity resolution is finite, i.e., all possible singularities can be resolved in a finite number of steps. Vladimir Ivanov does not discuss gaps in the papers of his precursors: he is concentrated on presenting a self-contained, comprehensive, and rigorous proof of the conjectures in question. First he follows the way passed by Gerrit Bol and Tilla Klotz, but soon 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).
The paper is written in a clear and friendly manner when ideas dominate over calculations and many general but not widely known results are included in the text. So, we can say that, despite of its length, the paper provides a reader with a real possibility of becoming confident that the index of an isolated umbilic point of an analytic surface is never greater than one and that on the sphere-homeomorphic surface there necessarily exist at least two umbilic points as it was conjectured by Constantin Carathéodory.