Umbilic points on closed convex surfaces (to be given in Russian)
by
Vladimir Ivanov
Sobolev Institute of Mathematics, Novosibirsk-90, 630090, Russia

According to the classical Caratheodory conjecture, there are at least two umbilic points on each closed convex smooth surface in the 3-dimensional Euclidean space. In the general case this problem remains open. One possible approach to its solution is related to another conjecture asserting that every domain on a smooth convex surface, whose spherical image contains a semisphere, contains an umbilic point. The main aim of the talk is to give a negative solution to the latter conjecture. Namely, we will discuss examples of analytic strictly convex surfaces S with the following properties: (i) S is arbitrary close to the sphere; (ii) all umbilic points of S are located at a domain of arbitrarily small diameter; (iii) spherical images of all umbilic points of S are located at a domain of arbitrarily small diamet

Date received: February 15, 2000