CR Embedding Research Articles

G-Equivariant Embedding Theorems for CR Manifolds of High Codimension

Let (X,T1,0X) be a (2n+1+d)-dimensional compact CR manifold with codimension d+1, d≥1, and let G be a d-dimensional compact Lie group with CR action on X and T be a globally defined vector field on X such that CTX=T1,0X⊕T0,1X⊕CT⊕Cg_, where g_ is the space of vector fields on X induced by the Lie algebra of G. In this work, we show that if X is strongly pseudoconvex in the direction of T and n≥2, then there exists a G-equivariant CR embedding of X into CN for some N∈N. We also establish a CR orbifold version of Boutet de Monvel’s embedding theorem.

Bergman–Szegő kernel asymptotics in weakly pseudoconvex finite type cases

Abstract We construct a pointwise Boutet de Monvel–Sjöstrand parametrix for the Szegő kernel of a weakly pseudoconvex three-dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman’s boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in ℂ 2 {\mathbb{C}^{2}} , in agreement with D’Angelo’s example. Finally, our results generalize a three-dimensional CR embedding theorem of Lempert.

CR embeddings of CR manifolds

We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type (n+ell ,k-ell ). While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension.

Left-invariant CR structures on 3-dimensional Lie groups

The systematic study of CR manifolds originated in two pioneering 1932 papers of Élie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on \(\mathrm {SU}_2= S^3\), deforming the standard ‘spherical’ structure. In this paper, mostly expository, we illustrate and clarify Cartan’s results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that \({\mathrm {SL}_2({\mathbb {R}})}\) admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of \({\mathrm {SL}_2({\mathbb {R}})}\) provide CR embedding or immersions of these structures. The same methods apply to all other 3-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for \(\mathrm {SU}_2\), the Heisenberg group, and the Euclidean group.

A stability theorem for projective CR manifolds

ABSTRACT We consider smooth deformations of the CR structure of a smooth 2-pseudoconcave compact CR submanifold of a reduced complex analytic variety outside the intersection with the support D of a Cartier divisor of a positive line bundle We show that nearby structures still admit projective CR embeddings. Special results are obtained under the additional assumptions that is a projective space or a Fano variety.

CR regular embeddings of 𝑆⁴ⁿ⁻¹ in ℂ²ⁿ⁺¹

Ahern and Rudin have given an explicit construction of a totally real embedding of S 3 S^3 in C 3 \mathbb {C}^3 . As a generalization of their example, we give an explicit example of a CR regular embedding of S 4 n − 1 S^{4n-1} in C 2 n + 1 \mathbb {C}^{2n+1} . Consequently, we show that the odd dimensional sphere S 2 m − 1 S^{2m-1} with m > 1 m>1 admits a CR regular embedding in C m + 1 \mathbb {C}^{m+1} if and only if m m is even.

CR Embedded Submanifolds of CR Manifolds

We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Cartan (or tractor) connections via a CR Gauss formula. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more naive methods. We conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.

An equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean space and CR regular embeddings into complex space

We show that a pseudo-holomorphic embedding of an almost-complex 2n -manifold into almost-complex (2n + 2) -Euclidean space exists if and only if there is a CR regular embedding of the 2n -manifold into complex (n + 1) -space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when n is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into \mathbb R^8 equipped with an almost-complex structure that need not be integrable.

CR regular embeddings and immersions of 6-manifolds into complex 4-space

We provide necessary and sufficient conditions in terms of characteristic classes for a closed smooth orientable 6-manifold to admit a CR regular immersion/embedding into C 4 \mathbb {C}^4 .

CR regular embeddings and immersions of compact orientable 4-manifolds into ℂ3

We show that a compact orientable 4-manifold M has a CR regular immersion into ℂ3 if and only if both its first Pontryagin class p1(M) and its Euler characteristic χ(M) vanish, and has a CR regular embedding into ℂ3 if and only if in addition the second Stiefel–Whitney class w2(M) vanishes.

On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions

It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding $f$ of a strictly psedoconvex hypersurface $M^{2n+1}\subset\bC^{n+1}$ into the sphere $\bS^{2N+1}\subset \bC^{N+1}$ is rigid, i.e. any other such local embedding is obtained from $f$ by composition by an automorphism of the target sphere $\bS^{2N+1}$, {\it provided} that the codimension $N-n<n/2$. In this paper, we consider the limit case $N-n=n/2$ in the simplest situation where $n=2$, i.e. we consider local CR embeddings $f\colon M^5\to \bS^7$. We show that there are at most two different local embeddings, up to composition with an automorphism of $\bS^7$. We also identify a subclass of 5-dimensional, strictly pseudoconvex hypersurfaces $M^5$ in terms of their CR curvatures such that rigidity holds for local CR embeddings $f\colon M^5\to \bS^7$.

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings f of a strictly pseudoconvex hypersurface M⊂Cn+1 into a sphere S in a higher dimensional complex space CN+1. The degeneracy of the mapping f will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d of the second fundamental form and all of its covariant derivatives is <n (here, n is the CR dimension of M), then f(M) is contained in a complex plane of dimension n+d+1. The converse of this statement is also true, as is easy to see. When the total rank d exceeds n, it is no longer true, in general, that f(M) is contained in a complex plane of dimension n+d+1, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n, then partial rigidity may still persist, but there is a “defect” k that arises from the ranks exceeding n such that f(M) is only contained in a complex plane of dimension n+d+k+1. Moreover, this defect occurs in general, as is illustrated by examples.

CR Embeddings and Kähler Manifolds with Pseudo-Conformally Flat Curvature Tensors

Under certain conditions on co-dimension and curvature tensors, the image of some CR or holomorphic maps are proved to be totally geodesic.

Rigidity of CR maps between Shilov boundaries of bounded symmetric domains

Our goal is to establish what seems to be the first rigidity result for CR embeddings between Shilov boundaries of bounded symmetric domains of higher rank. The result states that any such CR embedding is the standard linear embedding up to CR automorphisms. Our basic assumption extends precisely the well-known optimal bound for the rank one case. There are no other restrictions on the ranks, in particular, the difficult case when the target rank is larger than the source rank is also allowed.

Regularity for the CR vector bundle problem II

We derive a Ck+ 2 Holder estimate for Pφ, where P is either of the two solution operators in Henkin’s local homotopy formula for ∂b on a strongly pseudoconvex real hypersurface M in Cn , φ is a (0, q)-form of class Ck on M , and k ≥ 0 is an integer. We also derive a Ca estimate for Pφ, when φ is of class Ca and a ≥ 0 is a real number. These estimates require that M be of class Ck+ 2 , or Ca+2, respectively. The explicit bounds for the constants occurring in these estimates also considerably improve previously known such results. These estimates are then applied to the integrability problem for CR vector bundles to gain improved regularity. They also constitute a major ingredient in a forthcoming work of the authors on the local CR embedding problem. Mathematics Subject Classification (2010): 32V05 (primary); 32A26, 32T15 (secondary).

Regularity in the Local CR Embedding Problem

We consider a formally integrable, strictly pseudoconvex CR manifold M of hypersurface type, of dimension 2n−1≥7. Local CR, i.e., holomorphic, embeddings of M are known to exist from the works of Kuranishi and Akahori. We address the problem of regularity of the embedding in standard Hölder spaces C a(M), a∈R. If the structure of M is of class C m, m∈Z, 4≤m≤∞, we construct a local CR embedding near each point of M. This embedding is of class C a, for every a, 0≤a<m+(1/2). Our method is based on Henkin’s local homotopy formula for the embedded case, some very precise estimates for the solution operators in it, and a substantial modification of a previous Nash–Moser argument due to the second author.

Local embeddability of CR manifolds into spheres

In this paper we solve local CR embeddability problem of smooth CR manifolds into spheres under a certain nondegeneracy condition on the Chern–Moser’s curvature tensor. We state necessary and sufficient conditions for the existence of CR embeddings as finite number of equations and rank conditions on the Chern–Moser’s curvature tensors and their derivatives. We also discuss the rigidity of those embeddings.

Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

We consider holomorphic mappings sending a given Levi-nondegenerate pseudoconcave hypersurface M in C n + 1 into a nondegenerate hyperquadric of the same signature in PC N + 1 and show that if M is sufficiently close to a hyperquadric in a certain sense, then any two such mappings differ only by an automorphism of the hyperquadric.

Finite jet determination of CR embeddings

We prove finite jet determination results for smooth CR embeddings, which are of constant degeneracy, using the method of complete systems. As an application, we obtain a reflection principle for mappings between a Levinondegenerate hypersurface in ℂ N and a Levinondegenerate hypersurface in ℂN+1.We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric [14].

Symmetry of the tangential Cauchy-Riemann equations and scalar CR invariants

where φ and ψ are functions that are ∞ up to ∂ . ∂ inherits a geometric structure, called CR structure, from C +1 which is relevant for the biholomorphic equivalence of . Fefferman’s program, initiated in [5], is to compute all the scalar CR invariants of ∂ and to express the asymptotic expansion of φ modulo ( +2) and ψ modulo ( ) in terms of scalar CR invariants of ∂ . Fefferman’s invariant theory was developed further by T.N. Bailey, M.G. Eastwood, C.R. Graham, G. Komatsu and K. Hirachi, see [1], [7] and [8]. The main method is to obtain a defining function which is invariant under biholomorphic maps up to a power of determinants of biholomorphic maps and to construct a KahlerLorentz metric on a line bundle of which is invariant under local biholomorphic maps and unique modulo ( +1). In present paper our approach is viewing the CR invariants of a real hypersurface of C +1 as a scalar function defined on the jet space of CR embedding : → C +1 which is invariant under deformation of embedding. We express necessary and sufficient condition for scalar CR invariants using symmetry of the tangential CauchyRiemann equations. Let = { = 0} be a ∞ real hypersurface in C +1 and let { } =1 ... be a ∞ basis of the CR structure bundle 1 0( ) = C ( )∩ 1 0(C +1). A mapping = ( 1 . . . +1) : → C +1 is a CR embedding if