The CR umbilical locus of a real ellipsoid in $\mathbb{C}^2$

TL;DR

This paper characterizes the CR umbilical locus of a real ellipsoid in complex two-space, revealing it as a union of explicit curves and a complex variety defined by sextic equations, with explicit cases for symmetric ellipsoids.

Contribution

It provides a detailed description of the CR umbilical locus of real ellipsoids in a7a2, including explicit formulas and the structure of the locus.

Findings

The umbilical locus is the union of known curves and a non-trivial sextic variety.

Explicit descriptions are obtained for ellipsoids with certain symmetric semi-axes.

The locus includes a complex variety defined by two homogeneous sextic equations.

Abstract

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.