Hyperbolic Carathéodory conjecture

Details Hyperbolic Carathéodory conjecture Hyperbolic Carathéodory conjecture

2006-11-21

arxiv.org/abs/math/0611630v1

Mathematics

Differential Geometry

Latex 25 pages, 10 figures

Scientific paper

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 Carath\'{e}odory 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.

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