The CR umbilical locus of a real ellipsoid in C2

This paper concerns the CR umbilical locus of a real ellipsoid in C2, 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 sufficie...

Full description

Saved in:
Bibliographic Details
Published in:Journal of mathematical analysis and applications 2023-02, Vol.518 (1), Article 126701
Main Author: Son, Duong Ngoc
Format: Article
Language:English
Subjects:
Cartan invariant
CR manifold
Umbilical point
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:This paper concerns the CR umbilical locus of a real ellipsoid in C2, 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.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126701