Characterizing embeddable deformations of compact strictly pseudoconvex CR three-manifolds

A compact strictly pseudoconvex CR manifold of real dimension at least five is always globally CR embeddable in some complex Euclidean space, but in real dimension three non-embeddability is generic: the integrability condition is vacuous, abstract deformations of an embedded hypersurface exist in abundance, and most of them fail to embed in any $\mathbb{C}^N$. Curry and Ebenfelt (Math. Ann. 389, 2024) settled the question infinitesimally near the standard CR 3-sphere: modulo contact diffeomorphisms, the embeddable deformations form a Fréchet submanifold identified with the Burns–Epstein harmonic family, yielding a unique normal form. The fundamental problem they pose remains open in natural generality: give an intrinsic, effective characterization of which abstract deformations of a compact strictly pseudoconvex CR 3-manifold embedded in $\mathbb{C}^2$ give globally CR embeddable structures, and describe the local and global structure of the space of embeddable CR structures beyond a small neighborhood of the standard sphere.

Background

A CR (Cauchy–Riemann) manifold is a real manifold $M$ together with a distinguished complex subbundle $T^{1,0}$ of the complexified tangent bundle that plays the role of the holomorphic tangent bundle of a hypersurface in complex Euclidean space. In real dimension three, $T^{1,0}$ is a complex line bundle, and a CR structure can equivalently be given by a contact distribution $H\subset TM$ (a maximally non-integrable hyperplane field) carrying a complex structure $J$ on its fibers. A CR manifold is strictly pseudoconvex when the Levi form - the intrinsic quadratic form measuring how $T^{1,0}$ fails to be closed under commutators - is definite; this is the CR analogue of strong convexity and holds on boundaries of strictly pseudoconvex domains. A CR manifold is (globally) embeddable if it is CR diffeomorphic to a real submanifold of some $\mathbb{C}^N$, i.e. if its abstract CR structure is induced by ambient holomorphic coordinates; it is locally embeddable if every point has such a neighborhood.

The embeddability question has a sharp dimensional break. By work of Boutet de Monvel and Kohn, embeddability of a compact strictly pseudoconvex CR manifold is equivalent to a closed-range property of the tangential Cauchy–Riemann operator $\bar{\partial}_b$ (the operator induced on the boundary by the ambient $\bar{\partial}$), and in real dimension at least five every compact strictly pseudoconvex CR manifold is embeddable. In real dimension three the situation changes drastically: the formal integrability condition on $T^{1,0}$ - the condition that in higher dimensions prunes which abstract CR structures can come from $\mathbb{C}^n$ at all - is vacuous, so generic abstract deformations exist, and non-embeddability is the generic phenomenon (Rossi exhibited SU(2)-invariant non-embeddable structures on the 3-sphere; Nirenberg constructed a locally non-embeddable example; Burns and Epstein proved that a generic compact strictly pseudoconvex CR structure in dimension three fails to embed in every $\mathbb{C}^N$).

On the standard CR 3-sphere $S^3\subset\mathbb{C}^2$ (the boundary of the unit ball with its induced structure), deformations of the CR structure that preserve the standard contact distribution are parametrized by complex-valued functions - deformation tensors $\varphi$ with $|\varphi|_\infty<1$ - via the rule that the deformed holomorphic direction is spanned by $Z_1+\varphi\,\bar{Z}_1$, where $Z_1$ spans the standard $T^{1,0}$. Eliashberg showed that a CR structure on $S^3$ can embed in $\mathbb{C}^2$ only if its underlying contact structure is the standard one, so this parametrization covers the embeddable regime. Lempert's stability theorem states that a small abstract deformation of a compact strictly pseudoconvex hypersurface in $\mathbb{C}^2$ is embeddable in some $\mathbb{C}^N$ if and only if it is embeddable in $\mathbb{C}^2$; his theory of extremal holomorphic discs for the Kobayashi metric (the intrinsic pseudo-distance of complex analysis, generated by holomorphic maps of discs) and the associated circular representation explains the known embeddable families through nonnegativity of Fourier coefficients computed in $S^1$-invariant frames. Concretely, Burns and Epstein exhibited an explicit infinite-dimensional family of embeddable small deformations: deformation tensors whose spherical-harmonic components (the decomposition of functions on $S^3$ into restrictions of harmonic homogeneous polynomials of bidegree $(p,q)$ on $\mathbb{C}^2$) vanish for $q<p+4$. Epstein's relative index theory for Szegő projectors (the orthogonal projections onto CR-holomorphic function spaces) implies stability properties of the embeddable set, which in particular is closed in the $C^\infty$ topology. Cheng and Lee constructed a local slice for the action of the contact diffeomorphism group on the space of CR structures. Intrinsic criteria are only one-sided: a positive CR Yamabe invariant (a CR analogue of the conformal Yamabe invariant) together with a nonnegative CR Paneitz operator (a fourth-order, CR-invariant differential operator) implies embeddability (Chanillo–Chiu–Yang), and embeddability forces Paneitz nonnegativity (Takeuchi); neither yields a characterization.

Curry–Ebenfelt (Math. Ann. 389, 2024) settle the problem near the standard structure: the embeddable deformations of the standard CR 3-sphere form a smooth tame Fréchet submanifold of $C^\infty(S^3,\mathbb{C})$ near the origin (an infinite-dimensional submanifold of the space of smooth complex-valued functions, 'tame' in the sense used in Nash–Moser implicit-function arguments); in a modified Cheng–Lee slice the embeddable deformations are exactly the Burns–Epstein family with a reality condition along the critical diagonal $q=p+4$, which settles a 1990 question of Burns and Epstein beyond their genericity ($G_\delta$) statement; embeddable structures admit a normal form unique up to the automorphisms of the sphere; and every infinitesimally embeddable deformation generates a canonical embeddable family. All of these results are local to a sufficiently small neighborhood of the standard structure on $S^3$ and hold modulo contact diffeomorphisms.

Problem Statement

Which abstract deformations of a compact strictly pseudoconvex CR 3-manifold $M$ embedded as a hypersurface in $\mathbb{C}^2$ give CR structures that are globally CR embeddable - in the parametrization of the source, for which deformation tensors $\varphi$ (complex-valued functions with $|\varphi|_\infty<1$ determining the deformed holomorphic direction $Z_1+\varphi\,\bar{Z}_1$) is the deformed structure $(M,H,J_\varphi)$ embeddable, equivalently (near an embedded hypersurface, by Lempert's stability theorem) embeddable in $\mathbb{C}^2$? The problem, as posed by Curry–Ebenfelt, asks for a characterization of the embeddable deformations that is intrinsic and effective, and for the structure of the space of embeddable CR structures. The open problem is to extend the slice and normal-form description beyond a sufficiently small neighborhood of the standard CR 3-sphere (to finite, non-infinitesimal deformations), to find a decision criterion going beyond the known one-sided conditions (the sufficient CR Yamabe/Paneitz hypothesis and the necessary Paneitz nonnegativity), and to give the corresponding description for deformations of compact embeddable strictly pseudoconvex CR 3-manifolds other than the sphere, where Eliashberg's contact-topological obstruction and the absence of the circular-representation tool apply.

References

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