Hyperbolic Carathéodory conjecture

V. Ovsienko, S. Tabachnikov

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


A quadratic point on a surface in ℝP3 is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact nondegenerate hyperbolic surface is 8; the relation between this and the classic Carathéodory conjecture is similar to the relation between the six-vertex and the four-vertex theorems on plane curves. Examples of quartic perturbations of the standard hyperboloid confirm our conjecture. Our main result is a linearization and reformulation of the problem in the framework of the 2-dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing the Tresse-Wilczynski theorem.

Original languageEnglish (US)
Pages (from-to)178-193
Number of pages16
JournalProceedings of the Steklov Institute of Mathematics
Issue number1
StatePublished - Sep 2007

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


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