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Stable umbilical points on perturbations of the sphere in C^2

Banff International Research Station for Mathematical Innovation and Discovery

The standard CR structure on the three dimensional sphere can be deformed in such a way that the deformed structures have no (CR) umbilical points. A 1-parameter family of such deformations was essentially discovered by E.~Cartan (and later studied by Cap, Isaev, Jacobowitz). The CR manifolds in this family, however, cannot be embedded in $\mathbb C^2$. It is an open question whether the unit sphere can be perturbed in $\mathbb C^2$ such that no umbilical points remain on the perturbed CR manifolds. In this talk, we shall discuss an approach to this problem, and describe some recent results. One of the results that will be described guarantees stable (in a sense to be made precise in the talk) umbilical points on generic perturbations of ”almost circular type”. This complements a previous result by the speaker and Son Duong on existence of umbilical points on circular three-dimensional CR manifolds.

Mathematics; Several complex variables and analytic spaces; Potential theory; Global analysis

video/mp4

Author affiliation: University of California at San Diego

2017-02-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59652

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/