Hyperbolic Carath\'eodory Conjecture

S. L. Tabachnikov ; V. Yu. Ovsienko

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 1, Tome 258 (2007), pp. 185-200

Résumé

A quadratic point on a surface in $\mathbb R\mathrm P^3$ 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.