Nondegenerate CR Research Articles

On equivalence problem for 2–nondegenerate CR geometries with simple models

In this article, we solve the equivalence problem for 2–nondegenerate CR geometries that have (at every point) a homogeneous space G/H as a maximally symmetric model for G simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric models with a real simple CR automorphism group. In particular, we construct (local) embeddings of these models into complex space.

Higher order Levi forms on homogeneous CR manifolds

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous CR manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal than 3 and the compact ones less or equal 2. Finally we construct by Lie extensions weakly nondegenerate CR vector bundles with arbitrary orders of nondegeneracy.

On the Levi-flat Plateau problem

We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary $M$. We also study boundary regularity of CR automorphisms of domains in ${\mathbb C}^n \times {\mathbb R}$.

On the Pseudohermitian Curvature of Contact Semi-Riemannian Manifolds

Let M be a contact semi-Riemannian manifold, equivalently a non degenerate almost CR manifold. In this paper we study the pseudo-hermitian Ricci curvature, pseudo-Einstein and $$\eta $$-Einstein manifolds. Then, by using the pseudo-Einstein and the $$\eta $$-Einstein conditions, some rigidity theorems are established to characterize Sasakian manifolds among nondegenerate CR manifolds. In particular, if the Webster metric $$g_\theta $$ of nondegenerate CR structure $$({\mathcal {H}},\theta ,J)$$ is pseudo-Einstein with Webster scalar curvature $${\hat{r}}\ne 0$$, then there exists a real constant $$t\ne 0$$ for which the Webster metric associated to $$({\mathcal {H}},t\theta ,J)$$ is Einstein–Sasakian.

On the geometric order of totally nondegenerate CR manifolds

A CR manifold M, with CR distribution $$\mathcal {D}^{10} \subset T^\mathbb {C}M$$ , is called totally nondegenerate of depth $$\mu $$ if: (a) the complex tangent space $$T^\mathbb {C}M$$ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $$\mu $$ fields in $$\mathcal {D}^{10} + \overline{\mathcal {D}^{10}}$$ ; (b) for each integer $$2 \le k \le \mu -1$$ , the families of all vector fields that might be determined by iterated Lie brackets between at most k fields in $$\mathcal {D}^{10} + \overline{\mathcal {D}^{10}}$$ generate regular complex distributions; (c) the ranks of the distributions in (b) have the maximal values that can be obtained amongst all CR manifolds of the same CR dimension and satisfying (a) and (b)—this maximality property is the total nondegeneracy condition. In this paper, we prove that, for any Tanaka symbol $$\mathfrak {m}= \mathfrak {m}^{-\mu } + \cdots + \mathfrak {m}^{-1}$$ of a totally nondegenerate CR manifold of depth $$\mu \ge 4$$ , the full Tanaka prolongation of $$\mathfrak {m}$$ has trivial subspaces of degree $$k \ge 1$$ , i.e. it has the form $$\mathfrak {m}^{-\mu } + \cdots + \mathfrak {m}^{-1} + \mathfrak {g}^0$$ . This result has various consequences. For instance it implies that any (local) CR automorphism of a regular totally nondegenerate CR manifold is uniquely determined by its first order jet at a fixed point of the manifold. It also gives a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds.

Ultradifferentiable CR Manifolds

In this article, the notion of ultradifferentiable CR manifold is introduced and an ultradifferentiable regularity result for finitely nondegenerate CR mappings is proven. Here, ultradifferentiable means with respect to Denjoy–Carleman classes defined by weight sequences. Furthermore, the regularity of infinitesimal CR automorphisms on ultradifferentiable abstract CR manifolds is investigated.

Projective geometry of Sasaki–Einstein structures and their compactification

We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete non-compact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the K\"ahler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail.

Biholomorphic equivalence to totally nondegenerate model CR manifolds

Applying Elie Cartan’s classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka’s model of CR dimension one and codimension $$k> 1$$ , whence of real dimension $$2+k$$ , is reducible to some absolute parallelism, namely to an $$\{e\}$$ -structure on a certain prolonged manifold of real dimension either $$3+k$$ or $$4+k$$ . The proof relies upon a careful weight analysis on the structure equations associated with the mentioned problem of equivalence. As one of the applications of the achieved results, we also reconfirm in CR dimension one Beloshapka’s maximum conjecture on the holomorphic rigidity of his models in certain lengths equal or greater than three.

On Symmetric CR Geometries of Hypersurface Type

We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurface type are covered by CR geometries with a compatible pseudo-Riemannian metric preserved by all symmetries. We construct examples of simply connected flat non-degenerate symmetric CR geometries of hypersurface type that do not carry a pseudo-Riemannian metric compatible with the symmetries.

On the maximum conjecture

Abstract We verify the maximum conjecture on the rigidity of totally nondegenerate model CR manifolds in the following two cases: (i) for all models of CR dimension one, (ii) for the so-called full-models, namely those in which their associated symbol algebras are free CR. In particular, we discover that in each arbitrary CR dimension and length ≥ 3 {\geq 3} , there exists at least one totally nondegenerate model, enjoying this conjecture. Our proofs rely upon some recent results in the Tanaka theory of transitive prolongation of fundamental algebras.

On the Lichnerowicz conjecture for CR manifolds with mixed signature

We construct examples of nondegenerate CR manifolds with Levi form of signature (p,q), 2≤p≤q, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature (1,n−1) that is not locally CR flat and admits an essential CR vector field. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.

Flattening a non-degenerate CR singular point of real codimension two

This paper continues the previous studies in two papers of Huang–Yin [HY16,HY17] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in $${{\mathbb {C}}^{n+1}}$$ with n + 1 ≥ 3, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY16] and a formal theory approach used in [HY17], we are able to provide more or less a complete solution to the flattening problem for a non-degenerate CR singular point along the lines of such studies. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting.

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

Let $$M \subset {\mathbb {C}}^{n+1}$$ , $$n \ge 2$$ , be a real codimension two CR singular real analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real analytic function on M that is CR outside the CR singularities extends to a holomorphic function in a neighbourhood of M. Our motivation is to prove the following analogue of the Hartogs–Bochner theorem. Let $$\Omega \subset {\mathbb {C}}^n \times {\mathbb {R}}$$ , $$n \ge 2$$ , be a bounded domain with a connected real analytic boundary such that $$\partial \Omega $$ has only nondegenerate CR singularities. We prove that if $$f :\partial \Omega \rightarrow {\mathbb {C}}$$ is a real analytic function that is CR at CR points of $$\partial \Omega $$ , then f extends to a holomorphic function on a neighbourhood of $${\overline{\Omega }}$$ in $${\mathbb {C}}^n \times {\mathbb {C}}$$ .

Flattening of CR singular points and analyticity of the local hull of holomorphy II

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in Cn with n≥3. The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22]. We split [22] into two independent articles to avoid it being too long.

Extension of CR functions from boundaries in Cn x R

Let $\Omega \subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial \Omega$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial \Omega \to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial \Omega$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\bar{\Omega}$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial \Omega$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\bar{\Omega}$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial \Omega$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.

Flattening of CR singular points and analyticity of the local hull of holomorphy I

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in $${\mathbb C}^n$$ with $$n\ge 3$$ near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on the case of CR singular points with at least one elliptic direction. We solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting. The results in this paper and those in (Flattening of CR singular points and analyticity of the local hull of holomorphy II, p. 60, 2014) are taken from our arxiv post (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012). We split (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012) into two independent articles to avoid it being too long.

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Since Poincare’s celebrated paper [19] published in 1907, there has been a growing literature concerned with the equivalence problem for real submanifolds in complex space (see e.g., [4, 6, 7, 11, 13, 14, 22] for some recent works as well as the references therein). One interesting phenomenon, observed by Webster for biholomorphisms of Levi nondegenerate hypersurfaces [23], is that the biholomorphic equivalence of some types of real-algebraic submanifolds of a complex space implies their algebraic equivalence. In this paper, we show that this very phenomenon holds for algebraic deformations of germs of minimal holomorphically nondegenerate realalgebraic CR submanifolds in complex space. Let us recall that a germ of a real-algebraic CR submanifold (M,p) ⊂ (Cn, p) is minimal if there exists no proper CR submanifold N ⊂ M through p of the same CR dimension as M . It is holomorphically nondegenerate if there exists no nontrivial holomorphic vector field tangent to M near p (see [21]). An algebraic deformation of (M,p) is a real-algebraic family of germs at p of real-algebraic CR submanifolds (Ms, p)s∈Rk in Cn, defined for s ∈ Rk near 0, such that M0 = M . We say that two such deformations (Ms, p)s∈Rk and (Nt, p′)t∈Rk are biholomorphically equivalent if there exists a germ of a realanalytic diffeomorphism φ : (Rk, 0) → (Rk, 0) and a holomorphic submersion B : (Cz × Cu, (p, 0)) → (Cn, p′) such that z → B(z, s) is a biholomorphism sending (Ms, p) to (Nφ(s), p′) for all s ∈ Rk close to 0. We shall say that such

CR-tractors and the Fefferman space

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non-degenerate CR manifold of hypersurface type. In particular, we construct and treat the basic objects that relate the natural bundles and natural operators on the two spaces. This is illustrated with several applications: We prove that a number of conformally invariant overdetermined systems, including Killing form equations and the equations for twistor spinors, admit non-trivial solutions on any Fefferman space. We show that, on a Fefferman space, the space of infinitesimal conformal isometries naturally decomposes into a direct sum of subspaces, which admit an interpretation as solutions of certain CR invariant PDE's. Finally we explicitly analyse the relation between tractor calculus on a CR manifold and the complexified conformal tractor calculus on its Fefferman space, thus obtaining a powerful computational tool for working with the Fefferman construction.

Analytic stability of the CR cross-cap

For m < n, any real analytic m-submanifold of complex n-space with a nondegenerate CR singularity is shown to be locally equivalent, under a holomorphic coordinate change, to a fixed real algebraic variety defined by linear and quadratic polynomials. The situation is analogous to Whitney?s stability theorem for cross-cap singularities of smooth maps. The complex analyticity of the normalizing transformation is proved using a rapid convergence argument.

CR Weyl geometry and connections of the canonical bundle

A nondegenerate CR manifold has a natural conformal class consisting of hermitian forms, which are called Levi forms, on the hyperdistribution. It is important to study the objects on CR manifolds which are independent of the choice of Levi forms. A CR Einstein-Weyl structure defined in [8] is one of such objects. Our purpose is to find nondegenerate CR manifolds with CR Einstein-Weyl structures. In this paper, we shall obtain many examples of CR Einstein-Weyl manifolds. These examples are actually derived from the fact that CR Einstein-Weyl structures are closely concerned with connections of the canonical bundle of CR manifolds.