A new invariant equation for umbilical points on real hypersurfaces in $\mathbb{C}^2$ and applications

We introduce a new sequence of $\operatorname{CR}$ invariant determinants on a three-dimensional $\operatorname{CR}$ manifold $M$ embedded in $\mathbb{C}^2$ (which is a different and harder to handle case than $\mathbb{C}^n$ with $n \geq 3$). The lowest order invariant represents E. Cartan’s 6th order invariant (the umbilical “tensor”), whose zero locus yields the set of umbilical points on $M$, whenever $M$ is Levi-nondegenerate. Moreover, this invariant extends regularly to (and vanishes at) all Levi-degenerate points of $M$, implying e.g. real-analyticity (resp. real-algebraicity) of the umbilical set across such points whenever $M$ is real-analytic (resp. real-algebraic). As a further application, we show that generic, almost circular perturbations of the sphere always contain curves of umbilical points.

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The first author was supported in part by the NSF grant DMS-1600701.

Received 25 July 2016

Accepted 25 June 2017

Published 30 December 2019