Hyperbolic Carathéodory conjecture

Valentin Ovsienko (1) , Serge Tabachnikov (2)

1 ICJ - Institut Camille Jordan
2 PENN STATE - Penn State Math Dept

A quadratic point on a surface in $RP^3$ 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 non-degenerate 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 2-dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing Tresse-Wilczynski's theorem.

Domaines: Géométrie différentielle [math.DG]; Géométrie symplectique [math.SG]

Valentin Ovsienko, Serge Tabachnikov. Hyperbolic Carathéodory conjecture. 2006. ⟨hal-00115324⟩