CR analogue of Hamburger's theorem: at least two CR umbilical points on compact strongly convex real-analytic hypersurfaces in $\mathbb{C}^2$

For a real hypersurface $M\subset\mathbb{C}^2$, a CR umbilical point is a point where Cartan's sixth-order CR invariant vanishes, i.e., where a biholomorphic image of the unit sphere has contact with $M$ of order at least six. Huang and Ji proved that every real ellipsoid in $\mathbb{C}^2$ carries at least four CR umbilical points and posed two open questions: a CR analogue of Hamburger's classical umbilic-point theorem — must every compact strongly convex real-analytic hypersurface in $\mathbb{C}^2$ admit at least two CR umbilical points — and whether any embeddable compact three-dimensional CR manifold can have no CR umbilical points at all. The second question was resolved affirmatively by Ebenfelt–Son–Zaitsev (2016; published 2018), whose examples are non-convex three-torus boundaries; the strongly convex Hamburger-type question remains open: no general lower bound (not even one umbilical point) is known for this class, and no strongly convex counterexample is known. This task concerns precisely this surviving open core, stated as a prove-or-refute question.

Background

A real hypersurface $M\subset\mathbb{C}^{n+1}$ carries an intrinsic Cauchy–Riemann (CR) structure: at each point the complex tangent space $T^cM=T M\cap i\,T M$ has complex dimension $n$, called the CR dimension. The induced Hermitian form on $T^cM$ (the Levi form) is definite everywhere when $M$ is strictly pseudoconvex — the natural 'curvature-positive' class bounding pseudoconvex domains. A smooth compact hypersurface is strongly convex when it bounds a convex domain with definite second fundamental form, which implies strict pseudoconvexity but is strictly stronger and, unlike pseudoconvexity, is not preserved by biholomorphisms. Chern and Moser (1974) solved the local biholomorphic equivalence problem for such hypersurfaces: in suitable 'normal' coordinates a hypersurface is compared to the unit sphere (the flat model), and the failure of higher-order contact with the sphere is measured by curvature invariants whose vanishing at a point defines a CR umbilical point, in direct analogy with umbilic points of surfaces in $\mathbb{R}^3$ (points where the two principal curvatures coincide). In CR dimension $n\geq 2$ the obstruction is the fourth-order Chern–Moser tensor, and Webster proved that a generic real ellipsoid in $\mathbb{C}^{n}$ with $n\geq 3$ has no CR umbilical points at all. In $\mathbb{C}^2$ (CR dimension one) the fourth-order tensor vanishes identically and the leading obstruction is a sixth-order invariant discovered by É. Cartan: the coefficient $c_{2,4}$ of the Chern–Moser normal form, a CR-invariant function whose vanishing at $p$ means that some biholomorphic image of the sphere osculates $M$ at $p$ to order at least six. Its vanishing locus is the CR umbilical locus; Chern–Moser proved that the stability group at a non-umbilical point of a three-dimensional CR manifold has at most two elements. In this setting Huang and Ji proved that every real ellipsoid in $\mathbb{C}^2$ admits at least four CR umbilical points, exhibiting a sharp contrast with Webster's higher-dimensional nonexistence. Their result suggested a CR analogue of a classical question from surface geometry: the Carathéodory conjecture asserts that every compact surface bounding a convex body in $\mathbb{R}^3$ has at least two umbilic points, proved in the real-analytic case by Hamburger via a Poincaré–Hopf index argument for the principal-line foliation (whose local index bound, the Loewner conjecture, remains open in the smooth category). Huang and Ji asked whether every compact strongly convex real-analytic hypersurface in $\mathbb{C}^2$ likewise admits at least two CR umbilical points, and, companion-wise, whether an embeddable compact three-dimensional CR manifold without CR umbilical points exists. The companion question — the existence part of the original Chern–Moser problem — was answered by Ebenfelt–Son–Zaitsev: the boundary of the Reinhardt-type domain $\{(\log|z|)^2+(\log|w|)^2<\varepsilon^2\}\subset\mathbb{C}^2$ is compact, real-analytic, strictly pseudoconvex and nowhere umbilical (it is a three-torus, hence far from convex). Subsequent work found more nowhere-umbilical examples (boundaries of thin Grauert tubes, i.e., adapted-complex-structure tube neighborhoods of hyperbolic surfaces, studied by Foo–Merker–Ta), none of them convex; and it established forced existence of umbilics only under symmetry or near-sphericity hypotheses (complete circular domains, free transverse circle actions, almost-circular perturbations of the sphere, ellipsoids). What remains untouched is precisely the strongly convex class of the Hamburger analogue: neither a general existence theorem (even of a single CR umbilical point) nor a strongly convex counterexample is known.

Problem Statement

Let $M\subset\mathbb{C}^2$ be a compact strongly convex real-analytic hypersurface, i.e., the boundary of a bounded strongly convex real-analytic domain in $\mathbb{C}^2$. Recall that $p\in M$ is a CR umbilical point when Cartan's sixth-order CR invariant (the normal-form coefficient $c_{2,4}$, equivalently the unique complex-valued Cartan curvature function on the three-dimensional CR manifold $M$) vanishes at $p$, meaning that a biholomorphic image of the unit sphere osculates $M$ at $p$ to order at least six. Question, posed by Huang–Ji after their theorem that every ellipsoid in $\mathbb{C}^2$ has at least four CR umbilical points, as the CR analogue of Hamburger's theorem on convex surfaces in $\mathbb{R}^3$: must $M$ admit at least two CR umbilical points? Prove the affirmative bound for the whole class, or construct a compact strongly convex real-analytic hypersurface in $\mathbb{C}^2$ with at most one CR umbilical point. Note the current state: no general existence result (not even a single forced CR umbilical point) is known under strong convexity alone, while all known nowhere-umbilical compact examples abandon convexity; the companion question of Huang–Ji on embeddable manifolds without CR umbilical points was resolved by non-convex examples, so strong convexity is exactly the surviving hypothesis.

References

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