$\mathbb{C}^2$中实椭球的CR脐轨迹
The CR umbilical locus of a real ellipsoid in $\mathbb{C}^2$
摘要：
本文研究了实椭球在$\mathbb{C}^2$中的CR脐轨迹，即椭球与球面的6阶双全纯像密切的点集。Huang和Ji证明了这个轨迹是非空的。Ebenfelt和Zaitsev证明，对于足够“接近”球面的椭球，轨迹实际上包含一条“稳定”的脐点曲线。Foo，Merker和Ta后来给出了一条包含在其中的显式曲线，本文的主要结果表明脐轨迹是上述曲线和一个由两个四实变量齐次实六次方程定义的非平凡实簇的并.当存在合适的相等半轴对时，其中一个因子允许我们明确地确定轨迹。
英文摘要：
This paper concerns the CR umbilical locus of a real ellipsoid in $\mathbb{C}^2$, the set of points at which the ellipsoid can be osculated by a biholomorphic image of the sphere up to 6th order. Huang and Ji proved that this locus is non-empty. Ebenfelt and Zaitsev proved that, for ellipsoids that are sufficiently "closed" to the sphere, the locus actually contains a "stable" curve of umbilical points. Foo, Merker and Ta later provided an explicit curve that is contained in it. The main result in this paper exhibits that the umbilical locus is the union of the curves mentioned above and a non-trivial real variety which is defined by two homogeneous real sextic equations of four real variables. When there are suitable pairs of equal semi-axes, one of them factors allowing us to determine the locus explicitly.
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