A ball-like bounded strictly pseudoconvex domain in $\mathbb C^2$ without CR umbilical points (the refined Chern–Moser question)

In 2018 Ebenfelt, Son, and Zaitsev resolved the classical Chern–Moser question by constructing a bounded strictly pseudoconvex domain in $\mathbb C^2$ , diffeomorphic to $T^2\times\mathbb D$ , whose boundary is a compact strictly pseudoconvex CR manifold with no CR umbilical points, and they explicitly posed the natural refinement: can such a domain be chosen ball-like, i.e. with closure diffeomorphic to the closed unit ball in $\mathbb C^2$ , so that the boundary is diffeomorphic to $S^3$ ? The refinement remains open to date. The only known CR structures on $S^3$ without umbilical points (the Rossi spheres and their perturbations) are non-embeddable into $\mathbb C^2$ ; every known ball-like or symmetric class of strictly pseudoconvex boundaries (real ellipsoids, circular and $U(1)$ -symmetric hypersurfaces, generic almost-circular perturbations of the sphere) must carry umbilical points; and all subsequently constructed non-umbilical examples (the torus boundaries of the source paper and the boundaries of Grauert tubes over closed hyperbolic surfaces) have non-spherical topology. The problem asks for either an embedded ball-like example or a theorem forcing an umbilical point on every such boundary.

Background

A smooth real hypersurface $M\subset\mathbb C^2$ of real dimension three carries an induced CR structure (Cauchy–Riemann structure): the complex structure of the ambient space makes the complex tangent bundle $H=TM\cap iTM$ a complex line bundle over $M$ . The hypersurface (or a bounded domain it bounds) is strictly pseudoconvex where the Levi form — the Hermitian form on $H$ induced by $i\partial\bar\partial\rho$ of a smooth defining function $\rho$ — is positive definite; this is the basic strong positivity condition in several complex variables, and such boundaries have bases of holomorphically convex (Stein) neighborhoods and support an intrinsic biholomorphism-invariant geometry. In their 1974 theory of real hypersurfaces in complex manifolds, Chern and Moser introduced the CR umbilical points of such a hypersurface: points where the local CR curvature invariant vanishes and the hypersurface is, to the relevant order, osculated by the unit sphere. In complex dimension $n\geq 4$ a generic strictly pseudoconvex boundary has no umbilical points at all (a Thom transversality argument), and, by a result of Webster recounted in the source paper, every non-spherical real ellipsoid in $\mathbb C^n$ , $n\geq 3$ , is free of them; in $\mathbb C^2$ , however, umbilicity is detected by a sixth-order invariant going back to É. Cartan's classification work of the early 1930s — Cartan's umbilical tensor $Q$ — and the situation is genuinely different: Huang and Ji proved that every real ellipsoid in $\mathbb C^2$ must have umbilical points, and Chern and Moser's question of whether some bounded strictly pseudoconvex domain in $\mathbb C^2$ has a boundary with no CR umbilical points stayed open for four decades. Ebenfelt, Son, and Zaitsev settled it in 2018: for the domain $D_\epsilon=\{(\log|z|)^2+(\log|w|)^2<\epsilon^2\}$ , the boundary $M_\epsilon=\partial D_\epsilon$ is a compact strictly pseudoconvex CR manifold, locally homogeneous and without umbilical points. Their domain is diffeomorphic to $T^2\times\mathbb D$ , so $\partial D_\epsilon$ is a three-torus, and they explicitly asked whether the question can be reopened with the simplest possible topology: can $D$ be chosen with $\overline D$ diffeomorphic to the closed unit ball, so that $\partial D$ is diffeomorphic to $S^3$ ? Two structural facts frame this refinement. First, the only known compact strictly pseudoconvex CR structures on $S^3$ without umbilical points are the Rossi spheres $\mu^{(4)}_\alpha$ — a homogeneous family of deformations of the standard CR sphere introduced in Rossi's 1965 work — and their perturbations; these do not admit Stein neighborhoods at all, hence cannot occur as boundaries of domains in $\mathbb C^2$ , since an embedded compact strictly pseudoconvex hypersurface in $\mathbb C^2$ is the boundary of a bounded strictly pseudoconvex domain with a Stein neighborhood basis. Consequently, as the source paper observes using an extension theorem for local CR automorphisms and the classification of homogeneous three-dimensional CR manifolds, the boundary of any ball-like example cannot be locally homogeneous: a simply connected locally homogeneous example would be homogeneous, hence a Rossi sphere, hence not embeddable. Second, all known results in the opposite direction — every real ellipsoid in $\mathbb C^2$ has an umbilical point (Huang–Ji), every compact strictly pseudoconvex three-dimensional CR manifold with a free transverse circle action whose quotient is not a torus has an orbit of umbilical points, in particular every circular hypersurface in $\mathbb C^2$ (Ebenfelt–Son), and generic almost-circular perturbations of the unit sphere contain curves or surfaces of umbilical points (Ebenfelt–Zaitsev) — concern ball-like or highly symmetric classes, and none covers general ball-like domains. A later construction of Foo–Merker–Ta produces further non-umbilical examples (boundaries of thin Grauert tubes — canonical strictly pseudoconvex disk-bundle neighborhoods — over closed hyperbolic surfaces of genus at least two), but their topology is that of a circle bundle over a surface, never $S^3$ ; their paper records the sphere-like case as still open. Thus neither an embedded ball-like example without umbilics, nor any mechanism forcing an umbilical point on every embedded sphere-like strictly pseudoconvex boundary, is currently known.

Problem Statement

Does there exist a bounded strictly pseudoconvex domain $D\subset\mathbb C^2$ with smooth boundary such that (i) the closure $\overline D$ is diffeomorphic to the closed unit ball in $\mathbb C^2$ (which makes the boundary $\partial D$ diffeomorphic to the three-sphere $S^3$ ), and (ii) $\partial D$ has no CR umbilical points, i.e. Cartan's sixth-order umbilical tensor $Q$ is nonzero at every point of $\partial D$ ? This is the refinement of the classical Chern–Moser umbilical-point question explicitly proposed by Ebenfelt–Son–Zaitsev after their torus-boundary resolution.

References

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