Sharpness of the Ebenfelt–Minor two-embeddings bound: inequivalent local CR embeddings of strictly pseudoconvex hypersurfaces into spheres at critical codimension

A 2014 theorem of Ebenfelt and Minor shows that a smooth, connected, strictly pseudoconvex hypersurface $M\subset\mathbb{C}^{3}$ admits at most two non-constant CR mappings into the sphere $S^{7}\subset\mathbb{C}^{4}$ up to composition with an automorphism of $S^{7}$, and at most one such mapping for a curvature-identified subclass. The authors state that they know of no non-spherical example attaining the count of two in this critical codimension, and that nothing is known about the counting problem for general hypersurfaces in the borderline codimension regime $N-n+\mu(M)=n$ where the earlier rigidity theorem of Ebenfelt–Huang–Zaitsev no longer applies. According to [1] and the references therein, the sharpness question (construct a non-spherical $M\subset\mathbb{C}^{3}$ with CR complexity $\mu(M)=1$ admitting two inequivalent local CR embeddings into $S^{7}$, or prove that the bound is never attained) and the general borderline-codimension counting problem both remain open.

Background

A $CR$ hypersurface is the boundary-type real hypersurface $M^{2n+1}\subset\mathbb{C}^{n+1}$ carrying the induced Cauchy–Riemann structure: its complex tangent space at each point has complex dimension $n$, and a $CR$ mapping is a map whose differential respects the complex tangent directions (for the hypersurfaces considered here, a mapping that is the restriction of a holomorphic map of ambient neighborhoods). The Levi form of $M$ is the Hermitian form on the complex tangent bundle obtained by differentiating a defining function; $M$ is strictly pseudoconvex when this form is definite (positive, after orienting), which makes $M$ locally the boundary of a pseudoconvex domain. The unit sphere $S^{2N+1}\subset\mathbb{C}^{N+1}$ is the flat model of strictly pseudoconvex hypersurfaces, and its automorphism group (restrictions of the fractional-linear automorphisms of complex projective space $\mathbb{P}^{N+1}$ preserving the sphere, isomorphic to the projective group $PU(N+1,1)$) defines the natural equivalence under which CR mappings into the sphere are counted. A central theme since Webster and Chern–Moser is the rigidity or classification of transversal CR mappings sending a Levi-nondegenerate hypersurface $M\subset\mathbb{C}^{n+1}$ into the sphere or, more generally, a hyperquadric in higher-dimensional complex space. The CR (pseudoconformal) curvature introduced by Chern and Moser plays the role of the Weyl curvature of Riemannian geometry and vanishes exactly on locally spherical hypersurfaces, i.e. those locally biholomorphic to the sphere. The CR complexity $\mu(M)$, introduced by Ebenfelt and Shroff, is the minimal codimension $N_{0}-n$ of a target sphere $S^{2N_{0}+1}$ into which $M$ admits a local CR embedding; $\mu(M)=0$ exactly for locally spherical $M$, $\mu(M)=1$ for any non-spherical real ellipsoid in $\mathbb{C}^{3}$, and $\mu(M)$ may be infinite, since some strictly pseudoconvex hypersurfaces admit no local CR embedding into any sphere. The foundational rigidity theorem of Ebenfelt, Huang and Zaitsev states that if $\mu(M)=N_{0}-n$ and $f\colon M\to S^{2N+1}$ is a non-constant CR map with $N-n+\mu(M)<n$, then $f$ is obtained from a minimal embedding by a linear inclusion and an automorphism of the target sphere; in particular, in that range the embedding is unique up to equivalence. The inequality is sharp when $M$ is the sphere itself ($\mu(M)=0$): Faran showed there are four equivalence classes of maps $S^{3}\to S^{5}$, and the linear embedding and the Whitney map — a degree-two polynomial sphere map inequivalent to the linear one whenever $N-n\ge n$ — give two inequivalent maps $S^{2n+1}\to S^{2N+1}$ when $N-n=n$, which Huang and Ji proved are the only two. For a general (non-spherical) $M$, the borderline regime $N-n+\mu(M)=n$ was first attacked by Ebenfelt and Minor: in the simplest case $n=2$, i.e. $M^{5}\subset\mathbb{C}^{3}$ with $\mu(M)=1$ mapping into $S^{7}\subset\mathbb{C}^{4}$, they proved there are at most two non-constant CR mappings up to automorphisms of $S^{7}$, and they identified, via the rank and trace of the Chern–Moser curvature tensor acting as a linear map $L_{S}$ on symmetric $(2,0)$-tensors, a subclass of hypersurfaces for which there is at most one. Their proof studies the Gauss equation — the relation, for a transversal CR embedding $f\colon M\to S^{2N+1}$, between the CR curvature of $M$ and the CR second fundamental form of $f$ — directly counting its solutions rather than eliminating the curvature term as in rigidity proofs. The paper records that constructing inequivalent mappings in this critical codimension for non-spherical $M$ has proven elusive and that the authors have no example attaining the maximal count of two, while for the higher-dimensional borderline cases nothing at all is known for general $M$.

Problem Statement

Determine the number of local CR embeddings of a strictly pseudoconvex hypersurface into spheres in the borderline codimension regime $N-n+\mu(M)=n$ where the Ebenfelt–Huang–Zaitsev rigidity theorem no longer applies, with the sharpness of the Ebenfelt–Minor bound as its critical low-dimensional case. Concretely: (i) Does there exist a non-spherical, smooth, connected, strictly pseudoconvex hypersurface $M\subset\mathbb{C}^{3}$ with CR complexity $\mu(M)=1$ that admits two local CR embeddings into $S^{7}\subset\mathbb{C}^{4}$ inequivalent under any automorphism of $S^{7}$ — i.e. is the maximal count of two in Ebenfelt–Minor's theorem ever attained — or, conversely, does every such non-spherical $M$ admit at most one embedding up to equivalence (rigidity)? (ii) More generally, for a smooth, connected, strictly pseudoconvex hypersurface $M\subset\mathbb{C}^{n+1}$ with finite CR complexity and any $N\ge n+\mu(M)$ satisfying $N-n+\mu(M)=n$ with $n\ge 3$: determine whether the number of local CR embeddings $M\to S^{2N+1}$, up to automorphisms of the target sphere, is finite, and bound, count, or classify them. These two questions share one objective — completing the rigidity-versus-nonrigidity picture for CR embeddings into spheres exactly at the codimension threshold — with (i) the low-dimensional case and (ii) its higher-dimensional counterpart, the natural generalization that the source paper explicitly records as unexplored.

References

1. Peter Ebenfelt, André Minor. On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions. Transactions of the American Mathematical Society 366 (2014), 5693–5706. DOI: 10.1090/s0002-9947-2014-06085-6; arXiv:1208.0947. https://doi.org/10.1090/s0002-9947-2014-06085-6
2. Peter Ebenfelt, Xiaojun Huang, 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. https://doi.org/10.4310/cag.2004.v12.n3.a6
3. Xiaojun Huang, Shanyu Ji. Mapping $B^n$ into $B^{2n-1}$. Inventiones Mathematicae 145 (2001), 219–250. DOI: 10.1007/s002220100140. https://doi.org/10.1007/s002220100140
4. James J. Faran. Maps from the two-ball to the three-ball. Inventiones Mathematicae 68 (1982), 441–475. DOI: 10.1007/bf01389412. https://doi.org/10.1007/bf01389412
5. S. S. Chern, J. K. Moser. Real hypersurfaces in complex manifolds. Acta Mathematica 133 (1974), 219–271. DOI: 10.1007/BF02392146. https://doi.org/10.1007/BF02392146
6. Bernhard Lamel, Duong Ngoc Son. The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps. Annales de l'Institut Fourier 71 (2021), 2137–2167. DOI: 10.5802/aif.3438; arXiv:1907.00834. https://doi.org/10.5802/aif.3438
7. Peter Ebenfelt, Duong Son. On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres. Preprint (2012). arXiv:1205.1237. https://arxiv.org/abs/1205.1237
8. Ilya Kossovskiy, Ming Xiao. On the embeddability of real hypersurfaces into hyperquadrics. Advances in Mathematics 331 (2018), 239–267. DOI: 10.1016/j.aim.2018.04.001. https://doi.org/10.1016/j.aim.2018.04.001
9. Sean N. Curry, Peter Ebenfelt. Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds. Mathematische Annalen 389 (2024), 627–669. DOI: 10.1007/s00208-023-02658-y. https://doi.org/10.1007/s00208-023-02658-y
10. Sung-Yeon Kim, Dmitri Zaitsev. Rigidity of proper holomorphic maps between bounded symmetric domains. Mathematische Annalen 362 (2015), 639–677. DOI: 10.1007/s00208-014-1143-2. https://doi.org/10.1007/s00208-014-1143-2