论文标题
通过非旋转对称表面对Carathéodory猜想的研究
A Study of the Carathéodory Conjecture through Non-Rotationally Symmetric Surfaces
论文作者
论文摘要
Carathéodory的众所周知的猜想指出，在三维欧几里得空间中的每个足够光滑，闭合的凸表面都至少承认两个脐带。已经确定了所有旋转对称表面的猜想是正确的。在本文中，我们研究了两个没有旋转对称性的表面家族的脐带点，并计算其指数。特别是，我们发现形式的表面属于$ ax^{2k}+by^{2k}+cz^{2k} = 1 $，$ a，b，b，b，c> 0 $，$ k \ in \ mathbb {z} _ {> 1} _ {> 1} $ aff numbilic points：6 umbilic点：六个已知形式和另一种已知形式和另一个已知形式。对于$ a，b，c，k $的许多测试值，此类脐带分别具有$ -1/2 $和$ 1 $的指数。我们还探讨了脐带点对表面$ ax^2+am y+ay^2+εy^4+bz^2 = 1 $的参数$ε$的依赖性。特别是，对于$ a <b $和$ a> b，$恰好存在两个脐带点，索引1的$ε$比某些临界值小。对于较大的$ε，$ a> b $的$表面准确允许十点脐带点；对于$ a，b，ε，$的许多测试值，这些点具有索引1/2和-1。对于较大的$ε，$ a <b $的$表面接收十八个脐带；对于$ a，b，ε，$的许多测试值，这些点具有索引-1/2和1。
Carathéodory'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 $ε$ of the surface $ax^2+εx^4+ay^2+εy^4+bz^2=1$. In particular, for both $a<b$ and $a>b,$ there exist exactly two umbilic points with index 1 for $ε$ smaller than certain critical values. For larger $ε,$ surfaces with $a>b$ admit exactly ten umbilic points; for many tested values of $a,b,ε,$ these points have indices 1/2 and -1. For larger $ε,$ surfaces with $a<b$ admit eighteen umbilic points; for many tested values of $a,b,ε,$ these points have indices -1/2 and 1.
