Every real ellipsoid in ℂ² admits CR umbilical points

Abstract

Abstract. We prove that every real ellipsoid M ⊂ C 2 admits at least four umbilical points, which can be compared to the result of Webster that a generic real ellipsoid in C n with n ≥ 3 does not admit any umbilical point.

The umbilical locus on the boundary of strictly pseudoconvex domains in $\mathbb C^2$

“…Proof. This follows immediately from (12) and the corresponding transformation rule (well known and also readily verified by calculations similar to those in [7]) for J:…”

“…In particular, it follows that real ellipsoids close to the sphere always possess a curve of umbilical points, a fact previously proved (for all ellipsoids, not just those close to the sphere) by X. Huang and S. Ji in [12].…”

“…A simple Thom transversality argument (see, e.g., [7]) shows that a generic (i.e., sufficiently general) strictly pseudoconvex domain in C n with n ≥ 4 does not have any umbilical points in its boundary, and Webster [19] showed that, in particular, every non-spherical real ellipsoid in C n with n ≥ 3 has no umbilical points. On the other hand, in contrast with Webster's result and showcasing the point that the situation in C 2 and that in C n with n ≥ 3 is different, X. Huang and S. Ji [12] proved that every real ellipsoid in C 2 must have umbilical points. An application of Thom transversality in the case of strictly pseudoconvex domains Ω in C 2 shows that generically the set of umbilical points on the boundary M = ∂Ω is either empty (as in Theorem 1.1) or form smooth curves in M (as in Theorem 1.4 below).…”

A family of compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^2$ without umbilical points

“…A simple Thom transversality argument (see, e.g., [7]) shows that a generic (i.e., sufficiently general) strictly pseudoconvex domain in C n with n ≥ 4 does not have any umbilical points in its boundary, and Webster [17] showed that, in particular, every non-spherical real ellipsoid in C n with n ≥ 3 has no umbilical points. In contrast with Webster's result, and illustrating the point that the situation in C 2 and that in C n with n ≥ 3 is different, X. Huang and S. Ji [9] proved that every real ellipsoid in C 2 must have umbilical points. We mention here also two other recent papers, [6] and [7], in which the focus (regarding Chern-Moser's question) has been on proving results to the effect that certain classes of three-dimensional CR manifolds must possess umbilical points (supporting a possible 'no' as an answer to Chern-Moser's question).…”

Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in $\mathbb C^2$

“…Not much is known in general about this problem. It was shown by X. Huang and S. Ji [11] that every real ellipsoid in C 2 must have umbilical points. More recently, it was proved by the first author and S. Duong [6] that every circular M 3 ⊂ C 2 has umbilical points; in fact, it was proved in [6] that every compact three dimensional CR manifold with a transverse free CR U(1) (circle) action must have umbilical points provided that the Riemann surface M/U(1) has genus g = 1.…”