Intrinsic characterization of local CR embeddability into spheres in degenerate curvature regimes

Kim and Oh (2009) gave necessary and sufficient intrinsic conditions — finitely many equations and rank conditions on the Chern–Moser curvature tensor and its covariant derivatives — for a strictly pseudoconvex CR manifold of hypersurface type to admit a local CR embedding into a sphere $\mathbb S^{2N+1}\subset\mathbb C^{N+1}$, but only under a spectral nondegeneracy hypothesis on the curvature tensor, in a codimension range where that hypothesis is generic, and for CR dimension $n>1$. The open problem is to extend this to an intrinsic necessary and sufficient characterization of local CR embeddability into spheres in the remaining regimes: the spectrally degenerate case, codimensions beyond the genericity bound, and CR dimension one, where the curvature-based machinery degenerates.

Background

CR geometry studies real submanifolds of complex spaces through restrictions of the Cauchy–Riemann equations. A CR manifold of hypersurface type is a smooth $(2n+1)$-dimensional manifold $M$ together with a complex rank-$n$ subbundle $T^{1,0}M$ of its complexified tangent bundle that satisfies a Frobenius-type integrability condition; for a real hypersurface $M\subset\mathbb C^{n+1}$ this bundle is inherited from the ambient complex structure, and $n$ is called the CR dimension. The basic local invariant is the Levi form, the Hermitian form measuring non-tangential commutators of $(1,0)$ vector fields; $M$ is strictly pseudoconvex when this form is definite. A CR map is a smooth map whose differential sends $T^{1,0}M$ into the $(1,0)$-tangent bundle of the target, and a CR embedding of $M$ into the unit sphere $\mathbb S^{2N+1}\subset\mathbb C^{N+1}$ is a CR map that embeds $M$ as a CR submanifold of the sphere.

For strictly pseudoconvex hypersurfaces the sphere is the flat model: the Chern–Moser curvature tensor $S$ (Chern–Moser 1974) is a fundamental local invariant that vanishes exactly on germs locally CR equivalent to the sphere. Whether a given strictly pseudoconvex germ can be realized, even locally, inside a sphere of higher dimension is far more delicate. Forstnerič (1986) and Faran (1988) independently showed that a generic strictly pseudoconvex real-analytic hypersurface cannot be locally CR embedded into any sphere at all, while rigidity theorems (Webster 1979; Ebenfelt–Huang–Zaitsev 2004) show that when embeddings into sufficiently low-dimensional spheres exist they are essentially unique up to automorphisms of the target sphere. Kim and Oh (2006, 2009) attacked the local existence problem directly: encoding the prospective embedding as an overdetermined Pfaffian differential system on adapted frames (a system of first-order equations for differential one-forms on a bundle of frames, whose solvability is decided by Frobenius-type integrability conditions), they showed — under a spectral nondegeneracy hypothesis asserting, in essence, that suitable restricted curvature operators $\mathcal S_{V,W}$ associated with pairs of orthogonal complex subspaces $(V,W)$, $V\perp W$, $V\oplus W\neq T^{1,0}M$, achieve the maximal rank $N-n$ (the precise definition of the restricted operators $\mathcal S_{V,W}$ and the exact form of the hypothesis are given in their 2009 paper, which is the authoritative reference for both) — that local CR embeddability into $\mathbb S^{2N+1}$ is equivalent to finitely many explicit equations and rank conditions on $S$ and its covariant derivatives, with the embedding then essentially unique. The hypothesis is generic only in a limited range of codimensions $N-n\le\frac12\lfloor\frac{n-1}{2}\rfloor(\lfloor\frac{n-1}{2}\rfloor+1)$, and the arguments require $n>1$. Outside the reach of that theorem — when every admissible pair $(V,W)$ gives $\mathcal S_{V,W}$ smaller rank (the spectrally degenerate case), when the codimension exceeds the genericity bound and the theorem does not apply, or in CR dimension one, where the Chern–Moser tensor vanishes identically and local equivalence to the sphere is instead governed by Élie Cartan's classical curvature invariant for three-dimensional CR structures — no intrinsic necessary and sufficient conditions for local embeddability into spheres are known.

Problem Statement

Let $(M,p)$ be a germ of a smooth strictly pseudoconvex CR manifold of hypersurface type with CR dimension $n$, and let $N>n$ be an integer. Kim and Oh characterized local CR embeddability of $(M,p)$ into the sphere $\mathbb S^{2N+1}\subset\mathbb C^{N+1}$ by finitely many equations and rank conditions on the Chern–Moser curvature tensor and its covariant derivatives, but only under their spectral nondegeneracy hypothesis (existence of a pair $(V,W)$ of orthogonal complex subspaces with $V\oplus W\neq T^{1,0}M$ for which the restricted curvature operator $\mathcal S_{V,W}$ defined in that paper has rank $N-n$; the paper is the authoritative reference for the exact definition of $\mathcal S_{V,W}$ and the precise form of the hypothesis, of which this parenthetical is a summary), which is generic at most for $N-n\le\frac12\lfloor\frac{n-1}{2}\rfloor(\lfloor\frac{n-1}{2}\rfloor+1)$, and only for $n>1$. The problem is to give intrinsic necessary and sufficient conditions under which $(M,p)$ admits a local CR embedding into $\mathbb S^{2N+1}$, in the regimes not covered by that theorem: (i) the spectrally degenerate case, where no admissible pair $(V,W)$ makes $\mathcal S_{V,W}$ of rank $N-n$; (ii) all codimensions $N-n$ beyond the genericity bound; (iii) CR dimension $n=1$, where the Chern–Moser tensor carries no information and other curvature invariants must decide. Germs and target dimensions that satisfy every hypothesis of the Kim–Oh theorem are already characterized by it and are not part of the requested answer; in regime (ii) the request therefore concerns those germs to which the theorem does not already extend. A complete answer decides, for every germ and every codimension in these regimes that the Kim–Oh theorem does not already characterize, whether an embedding exists; combined with the existing rigidity theory this would complete the local existence-and-uniqueness theory of CR embeddings of strictly pseudoconvex germs into spheres. The global problem of proper holomorphic embedding of a strictly pseudoconvex domain into the ball extending a given boundary embedding (i.e. existence of a complex submanifold $X$ with $D\cong X\cap\mathbb B^{N+1}$) is a distinct, adjacent open problem and is not part of the requested answer.

References

1. Sung-Yeon Kim and Jong-Won Oh, Local embeddability of CR manifolds into spheres, Mathematische Annalen 344 (2009) 185–211, DOI 10.1007/s00208-008-0302-8, https://doi.org/10.1007/s00208-008-0302-8
2. Sung-Yeon Kim and Jong-Won Oh, Local embeddability of pseudohermitian manifolds into spheres, Mathematische Annalen 334 (2006) 783–807, DOI 10.1007/s00208-005-0710-y, https://doi.org/10.1007/s00208-005-0710-y
3. S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Mathematica 133 (1974) 219–271, DOI 10.1007/BF02392146, https://doi.org/10.1007/BF02392146
4. S. M. Webster, The rigidity of C-R hypersurfaces in a sphere, Indiana University Mathematics Journal 28 (1979) 405–416, DOI 10.1512/iumj.1979.28.28027, https://doi.org/10.1512/iumj.1979.28.28027
5. Peter Ebenfelt, Xiaojun Huang, and Dmitri Zaitsev, Rigidity of CR-immersions into spheres, Communications in Analysis and Geometry 12 (2004) 631–670, DOI 10.4310/CAG.2004.v12.n3.a6, arXiv:math/0206152, https://arxiv.org/abs/math/0206152
6. F. Forstnerič, Embedding strictly pseudoconvex domains into balls, Transactions of the American Mathematical Society 295 (1986) 347–368, DOI 10.1090/S0002-9947-1986-0831203-7, https://doi.org/10.1090/S0002-9947-1986-0831203-7
7. J. J. Faran, The nonimbeddability of real hypersurfaces in spheres, Proceedings of the American Mathematical Society 103 (1988) 902–904, DOI 10.1090/S0002-9939-1988-0947678-6, https://doi.org/10.1090/S0002-9939-1988-0947678-6
8. Sung-Yeon Kim, Regularity of rigid CR hypersurfaces in a sphere, Proceedings of the American Mathematical Society 139 (2011), no. 1, 129–137, DOI 10.1090/S0002-9939-2010-10411-2, https://doi.org/10.1090/S0002-9939-2010-10411-2
9. Peter Ebenfelt and André Minor, On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions, Transactions of the American Mathematical Society 366 (2014), no. 11, 5693–5706, DOI 10.1090/S0002-9947-2014-06085-6, arXiv:1208.0947, https://arxiv.org/abs/1208.0947
10. Peter Ebenfelt, Partial rigidity of degenerate CR embeddings into spheres, arXiv:1208.2731 (2012), https://arxiv.org/abs/1208.2731
11. Peter Ebenfelt and Duong Ngoc Son, On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres, arXiv:1205.1237 (2012), https://arxiv.org/abs/1205.1237
12. Xiaojun Huang and Dmitri Zaitsev, Non-embeddable real algebraic hypersurfaces, arXiv:1303.2385 (2013), https://arxiv.org/abs/1303.2385
13. Xiaojun Huang, Xiaoshan Li, and Ming Xiao, Non-embeddability into a fixed sphere for a family of compact real algebraic hypersurfaces, arXiv:1405.0778 (2014), https://arxiv.org/abs/1405.0778
14. Xiaojun Huang and Ming Xiao, Chern–Moser–Weyl tensor and embeddings into hyperquadrics, arXiv:1606.09145 (2016), https://arxiv.org/abs/1606.09145
15. Ilya Kossovskiy and Ming Xiao, On the embeddability of real hypersurfaces into hyperquadrics, arXiv:1509.01962 (2015), https://arxiv.org/abs/1509.01962
16. Xiaojun Huang, Shanyu Ji, and Brandon Lee, CR and holomorphic embeddings and pseudo-conformally flat metrics, arXiv:1210.4046 (2012), https://arxiv.org/abs/1210.4046
17. Hendrik Herrmann, Chin-Yu Hsiao, and Bernhard Lamel, Approximation of pseudohermitian structures via embeddings into spheres, arXiv:2506.14033 (2025), https://arxiv.org/abs/2506.14033
18. Hung-Lin Chiu, The fundamental and rigidity theorems for pseudohermitian submanifolds in the Heisenberg groups, Journal of Mathematical Study 54 (2021), no. 1, 1–27, DOI 10.4208/JMS.V54N1.21.01, arXiv:1802.04701, https://arxiv.org/abs/1802.04701