A Study of the Carath\'eodory Conjecture through Non-Rotationally Symmetric Surfaces

TL;DR

This paper investigates umbilic points on non-rotationally symmetric convex surfaces, providing explicit examples and analyzing how these points and their indices vary with surface parameters, thus contributing to understanding Carathéodory's conjecture.

Contribution

The study extends the analysis of umbilic points to specific non-rotationally symmetric surfaces, offering explicit computations and insights into their distribution and indices.

Findings

Surfaces of the form $ax^{2k}+by^{2k}+cz^{2k}=1$ have 14 umbilic points with indices -1/2 and 1.

For the surface $ax^2+ ext{epsilon} x^4 + ay^2 + ext{epsilon} y^4 + bz^2=1$, the number and indices of umbilic points depend on epsilon and parameters a, b.

Existence of exactly two umbilic points with index 1 for small epsilon, with the number and indices changing as epsilon increases.

Abstract

Carath\'eodory's well-known conjecture states that every sufficiently smooth, closed convex surface in three dimensional Euclidean space admits at least two umbilic points. It has been established that the conjecture is true for all rotationally symmetric surfaces; in this paper, we investigate the umbilic points of two families of surfaces without rotational symmetry, and compute their indices. In particular, we find that the family of surfaces of the form $ax^{2k}+by^{2k}+cz^{2k}=1$ with $a,b,c>0$, $k\in\mathbb{Z}_{>1}$ admit 14 umbilic points: six of one known form and eight of another. For many tested values of $a,b,c,k$, such umbilic points have indices $-1/2$ and $1$, respectively. We also explore the dependence of the umbilic points on the parameter $\epsilon$ of the surface $ax^2+\epsilon x^4+ay^2+\epsilon y^4+bz^2=1$. In particular, for both $a<b$ and $a>b,$ there exist exactly two umbilic points with index 1 for $\epsilon$ smaller than certain critical values. For larger $\epsilon,$ surfaces with $a>b$ admit exactly ten umbilic points; for many tested values of $a,b,\epsilon,$ these points have indices 1/2 and -1. For larger $\epsilon,$ surfaces with $a<b$ admit eighteen umbilic points; for many tested values of $a,b,\epsilon,$ these points have indices -1/2 and 1.