双曲Carathéodory猜想
Hyperbolic Carathéodory conjecture
摘要：
在$RP^3$中，曲面上的二次点是曲面可以异常良好地近似（直到3阶）的点。我们猜想一般紧致非退化双曲曲面上二次点的最少个数为8，这与经典的Carathéodory猜想之间的关系类似于平面曲线上的六顶点定理与四顶点定理之间的关系.标准双曲面的四次扰动的例子证实了我们的猜想。我们的主要结果是一个线性化和重新制定的问题的框架内的2维Sturm理论，我们还定义了一个签名的二次点和计算当地的正规形式恢复和推广Tresse-Wilczynski定理。
英文摘要：
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.
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