Convergence of Chern–Moser-type normal forms for real-analytic non-integrable almost CR structures

The classical Chern–Moser theory realizes every real-analytic Levi-nondegenerate hypersurface-type CR structure by a convergent normal form, and this convergent realization underlies the analytic local classification of such structures. Zaitsev (Amer. J. Math. 2012) extended the Chern–Moser normal form to non-integrable Levi-nondegenerate (hypersurface-type) almost CR structures in two constructions - an intrinsic normalization of the structure itself and an extrinsic one via 'quasi-CR' embeddings into complex space - and proved uniqueness (canonicity) up to the expected finite-dimensional parameters, together with 2-jet determination of CR-diffeomorphisms. However, both normal forms exist only as formal power series, and they are different in general: specializing to integrable CR structures, the intrinsic normal form differs from the extrinsic Chern–Moser normal form, and the two constructions coincide only for almost-complex structures (CR codimension zero). The open problem is whether, for real-analytic almost CR structures, these formal normal forms converge - equivalently, whether there exists a canonical convergent Chern–Moser-type normal form for non-integrable almost CR structures.

Background

A CR (Cauchy–Riemann) structure on a real manifold $M$ is a complex structure on a subbundle $H\subset TM$: an endomorphism $J:H\to H$ with $J^2=-\mathrm{id}$ whose $i$-eigenspace $H^{1,0}\subset\mathbb{C}\otimes TM$ satisfies the integrability condition $[H^{1,0},H^{1,0}]\subset H^{1,0}$. Dropping this integrability condition gives an almost CR structure; such structures arise naturally when deforming or gluing CR structures and as real submanifolds of almost complex manifolds (an almost complex structure is the special case $H=TM$; 'hypersurface type' means $H$ has corank one, as for a real hypersurface in $\mathbb{C}^{n+1}$). The Levi form is the bilinear form measuring the failure of $H$ to be closed under brackets modulo $H$; Levi nondegeneracy is the basic nondegeneracy assumption of the local theory. In the non-integrable case its role is shared with the non-integrability (Nijenhuis) tensor $N$, measuring the failure of $[H^{1,0},H^{1,0}]\subset H^{1,0}$: in coordinates it appears at the same differential order as the Levi form and enters all transformation formulas alongside it.

For an integrable real-analytic Levi-nondegenerate hypersurface-type CR structure, the theorem of Chern and Moser (1974) provides a normal form: locally, after a convergent biholomorphic change of ambient coordinates, the structure is realized as a real-analytic hypersurface $\operatorname{Im} w=\sum_j\varepsilon_j|z_j|^2+\sum_{k,l\ge 2}F_{kl}(z,\bar z,\operatorname{Re} w)$ with bihomogeneous terms $F_{kl}$ satisfying explicit trace conditions ($\operatorname{tr}F_{22}\equiv 0$, $\operatorname{tr}^2F_{23}\equiv 0$, $\operatorname{tr}^3F_{33}\equiv 0$, where $\operatorname{tr}:=\sum_j\varepsilon_j\partial^2/\partial z_j\partial\bar z_j$). This normal form is canonical up to the automorphism group of the hyperquadric $\operatorname{Im} w=\sum_j\varepsilon_j|z_j|^2$, and its convergence for real-analytic data - the fact that the normalizing transformation is an actual biholomorphism - is what turns it into an effective analytic classification tool. For merely smooth structures the same normalization exists only at the level of formal power series.

Zaitsev (Amer. J. Math. 134 (2012) 915–947) extended this normal form to non-integrable almost CR structures. Because a non-integrable almost CR structure admits no realization as a hypersurface in $\mathbb{C}^{n+1}$ even formally, he gave two constructions: an intrinsic one, imposing Chern–Moser-type normalization conditions directly on the map $L$ whose graph defines $H^{1,0}$ (via its 'radial part' - the evaluation of $L$ along the Euler vector field), and an extrinsic one, embedding the manifold into $\mathbb{C}^{n+1}$ by a quasi-CR embedding, i.e. an embedding that is not CR but is 'maximally CR' in that the induced structure agrees with the given one along the Euler (radial) direction. A key obstacle he highlights is that the Nijenhuis tensor appears at the same order as the Levi form, making the transformation formulas non-decoupled and ruling out the quadric approximation on which Chern–Moser theory rests. The paper proves: both normal forms exist as formal normal forms for smooth strongly nondegenerate structures (for positive-definite Levi form, strong nondegeneracy is automatic; in mixed signature it additionally requires nondegeneracy of a fixed linear combination of the Levi form with the transversal component of the Nijenhuis tensor); each is unique once an extended adapted frame is fixed (an adapted frame is a basis of $H_pM$ with a transversal vector in which the Levi form is diagonalized, extended by a 2-jet class of curves), so canonicity holds with the same parameter freedom as Chern–Moser; and applications to the equivalence problem - a complete system determining the 3-jet of a CR-diffeomorphism from its 2-jet, 2-jet determination of CR-diffeomorphisms, and a Lie group structure on automorphism groups. The two normal forms are in general different: specializing the intrinsic construction to integrable CR structures yields an intrinsic normal form different from the (extrinsic) Chern–Moser normal form of the same structure, and Remark 6.3 of the paper gives an explicit convergent example - the hypersurface $\operatorname{Im} w=z\bar z+u z^4\bar z^4$ - which is in the extrinsic normal form but does not satisfy the intrinsic normalization; the two constructions coincide only for almost-complex structures (CR codimension zero). A companion announcement (Ill. J. Math. 56 (2012) 273–280) proposes a unified normal form covering all Levi-nondegenerate structures, removing the extra hypothesis in mixed signature.

What the published theory does not address is convergence: for a real-analytic almost CR structure, no proof is given that the formal normalizing transformation, or the formal quasi-CR embedding, is induced by a convergent real-analytic map, and no counterexample to convergence is known. In the integrable case convergence is classical, so the analytic (convergent) counterpart of the almost CR normal form is precisely the missing part of the theory.

Problem Statement

Let $(M,H,J)$ be a germ at a point of a real-analytic hypersurface-type almost CR structure whose Levi form is nondegenerate (in mixed signature, assume in addition Zaitsev's strong nondegeneracy - nondegeneracy of the fixed linear combination $6iL+N^w$ of the Levi form with the transversal component of the Nijenhuis tensor). This is the class treated in the Amer. J. Math. paper; the Illinois J. Math. announcement extends the formal construction to all Levi-nondegenerate structures, but the class considered here is the strongly nondegenerate one. Zaitsev's theory produces, uniquely once an extended adapted frame is fixed, a formal Chern–Moser-type normal form in two ways: an intrinsic normalization of the map defining $H^{1,0}$, and a formal quasi-CR embedding into $\mathbb{C}^{n+1}$ satisfying the Chern–Moser normalization conditions.

The problem is the convergence of this formal theory:

1. Does the formal normalizing coordinate change converge - that is, is it induced by an actual real-analytic local diffeomorphism of $M$, so that the normal-form structure is itself real-analytic? Equivalently, does every real-analytic Levi-nondegenerate (resp. strongly nondegenerate) hypersurface-type almost CR structure admit a canonical convergent Chern–Moser-type normal form, with uniqueness in the sense of Theorem 1.4 of the source preserved?

2. In the extrinsic construction, does there exist an actual convergent quasi-CR embedding into $\mathbb{C}^{n+1}$ realizing the normal form (rather than only a formal one)?

(The intrinsic and extrinsic normal forms are known to be different in general - Remark 6.3 of the source exhibits an explicit integrable real-analytic example that is in the extrinsic normal form but fails the intrinsic normalization, and the two constructions coincide only for almost-complex structures, i.e. CR codimension zero - so Questions 1 and 2 concern two genuinely distinct normal-form constructions, and their convergence, if it holds, would not make the two normal forms coincide.)

References

1. 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
2. Dmitri Zaitsev, 'Normal forms for nonintegrable almost CR structures', American Journal of Mathematics 134 (2012), no. 4, 915–947. DOI: 10.1353/ajm.2012.0027, https://doi.org/10.1353/ajm.2012.0027 (arXiv preprint arXiv:0812.1104, https://arxiv.org/abs/0812.1104, carries the variant title 'Normal forms for almost non-integrable CR structures')
3. Dmitri Zaitsev, 'A normal form for all Levi-nondegenerate almost CR structures', Illinois Journal of Mathematics 56 (2012), no. 1, 273–280. DOI: 10.1215/ijm/1380287473, https://doi.org/10.1215/ijm/1380287473
4. Martin Kolář, Ilya Kossovskiy, and Dmitri Zaitsev, 'Normal forms in Cauchy–Riemann geometry', in: Analysis and Geometry in Several Complex Variables, Contemporary Mathematics 681, American Mathematical Society (2017), 153–177. DOI: 10.1090/conm/681/13685, https://doi.org/10.1090/conm/681/13685 (arXiv preprint arXiv:1606.08091, https://arxiv.org/abs/1606.08091, carries the variant title 'Normal forms in Cauchy-Riemann Geometry: a survey')
5. Ilya Kossovskiy and Dmitri Zaitsev, 'Convergent normal form for real hypersurfaces at a generic Levi-degeneracy', Journal für die reine und angewandte Mathematik 749 (2019), 201–225. DOI: 10.1515/crelle-2016-0034, https://doi.org/10.1515/crelle-2016-0034
6. M. Salah Baouendi, Linda Preiss Rothschild, Jörg Winkelmann, and Dmitri Zaitsev, 'Lie group structures on groups of diffeomorphisms and applications to CR manifolds', Annales de l'Institut Fourier 54 (2004), no. 5, 1279–1303. DOI: 10.5802/aif.2050, https://doi.org/10.5802/aif.2050