CR Manifolds Research Articles

1-Quasiconformal mappings and CR mappings on Goursat groups.

We show that 1-quasiconformal mappings on Goursat groups are CR or anti-CR mappings. This can reduce the determination of 1-quasiconformal mappings to the determination of CR automorphisms of CR manifolds, which is a fundamental problem in the theory of several complex variables.

The group of contact diffeomorphisms for compact contact manifolds

For a compact contact manifold it is shown that the anisotropic Folland-Stein function spaces form an algebra. The notion of anisotropic regularity is extended to define the space of Folland-Stein contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These results are used in a subsequent paper to analyze the action of the group of contact diffeomorphisms on the space of CR structures on a compact, three dimensional manifold.

Q-prime curvature on CR manifolds

Q-prime curvature, which was introduced by J. Case and P. Yang, is a local invariant of pseudo-hermitian structure on CR manifolds that can be defined only when the Q-curvature vanishes identically. It is considered as a secondary invariant on CR manifolds and, in 3-dimensions, its integral agrees with the Burns–Epstein invariant, a Chern–Simons type invariant in CR geometry. We give an ambient metric construction of the Q-prime curvature and study its basic properties. In particular, we show that, for the boundary of a strictly pseudoconvex domain in a Stein manifold, the integral of the Q-prime curvature is a global CR invariant, which generalizes the Burns–Epstein invariant to higher dimensions.

Sub-Laplacian Eigenvalue Bounds on CR Manifolds

We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjöstrand asymptotic law, and can be viewed as a CR version of Korevaar's bounds for Laplace eigenvalues of conformal metrics.

An Obata-type theorem in CR geometry

We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudohermitian manifold of dimension $2m + 1 \geq 5$. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. For this purpose, we establish an Obata-type theorem in CR geometry that characterizes the CR sphere in terms of a nonzero function satisfying a certain overdetermined system. Similar results are proved in dimension 3 under an additional condition.

On the Ampleness of Positive CR Line Bundles over Levi-flat Manifolds

We give an example of a compact Levi-flat CR 3-manifold with a positive-along-leaves CR line bundle which is not ample with respect to transversely infinitely differentiable CR sections. This example shows that we cannot improve the regularity of the Kodaira type embedding theorem for compact Levi-flat CR manifolds obtained by Ohsawa and Sibony.

A note on CR quaternionic maps

Abstract We introduce the notion of CR quaternionic map and we prove that any such realanalytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold M of dimension 4k-1 of a quaternionic manifold N such that TM generates a quaternionic subbundle of TN|M of (real) rank 4k, there exists, locally, a quaternionic submanifold of N containing M as a hypersurface.

ASYMPTOTIC APPROXIMATIONS AND A BOREL-TYPE RESULT FOR CR FUNCTIONS

We show that for any smooth CR manifold which has a peak function (in a weak sense) at some point p, formal power series at p can be approximated asymptotically by continuous CR functions. Furthermore, if the peak function satisfies a certain growth property, the asymptotic approximation is actually smooth. This in fact allows to invert, in a Borel-type theorem, the natural map taking a smooth CR function to its formal Taylor series.

Nonconstant CR morphisms betweenc compact strongly pseudoconvex CR manifolds and étale covering between resolutions of isolated singularities

Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities. We prove that any non- constant CR morphism between two $(2n−1)$-dimensional strongly pseudoconvex CR manifolds lying in an $n$-dimensional Stein variety with isolated singularities are necessarily a CR biholomorphism. As a corollary, we prove that any nonconstant self map of $(2n − 1)$-dimensional strongly pseudoconvex CR manifold is a CR automorphism. We also prove that a finite étale covering map between two resolutions of isolated normal singularities must be an isomorphism.

Asymptotics of ACH-Einstein Metrics

We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equation if we allow logarithmic terms, and that a local CR-invariant tensor arises as the obstruction to the existence of a log-free solution. Some properties of this new CR invariant, the CR obstruction tensor, are discussed.

Existence Results for the Prescribed Webster Scalar Curvature on Higher Dimensional CR Manifolds

Abstract This paper is about prescribing the Webster scalar curvature on a compact CR manifold of dimension 2n+1 ≥ 5, which is locally CR equivalent to the standard CR unit sphere S2n+1. The associated variational problem being noncompact we approach this issue with techniques from the critical points at infinity theory, combined with topological tools. Our existence results are based on a generalized Bahri-Coron type criteria. We also give an upper bound on the Morse index of the obtained solutions.

GCR-LIGHTLIKE SUBMANIFOLDS OF INDEFINITE NEARLY KAEHLER MANIFOLDS

We introduce CR, SCR and GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds and obtain their existence in indefinite nearly Kaehler manifolds of constant holomorphic sectional curvature <TEX>$c$</TEX> and of constant type <TEX>${\alpha}$</TEX>. We also prove characterization theorems on the existence of totally umbilical and minimal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds.

Subconformal Yamabe Equation and Automorphism Groups of Almost CR Manifolds

We construct a new curvature invariant for a subclass of contact sub-Riemannian structures which is transformed by a Yamabe type equation under a subconformal change. We apply this equation to characterize strongly pseudoconvex almost CR manifolds with nonproperly acting automorphism group. This generalizes Schoen’s theorem for the CR case in Schoen (Geom. Funct. Anal. 5:464–481, 1995).

Variétés CR polarisées et G-polarisées, partie I

Polarized and $G$-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\Cal F$ (given by the action of a Lie group $G$ in the $G$-polarized case) and a transverse CR distribution $(E,J)$. Polarized means that $(E,J)$ is roughly speaking invariant by $\Cal F$. Both structures are therefore linked up. The interplay between them gives to polarized CR-manifolds a very rich geometry. In this paper, we study the properties of polarized and $G$-polarized manifolds, putting special emphasis on their deformations.

The Equivalence Problem for Five-dimensional Levi Degenerate CR Manifolds

The Equivalence Problem for Five-dimensional Levi Degenerate CR Manifolds

CR immersions and Lorentzian geometry

Using tools from Lorentzian geometry (arising from the presence of the Fefferman metric) we prove a Takahashi type theorem (for a class of pseudohermitian immersionscoveredbyconnection-preservingequivariantimmersionsamongthetotal spacesofthecanonicalcirclebundles)thusrelatingthegeometryofapseudohermitian immersion from a strictly pseudoconvex CR manifold M into an odd dimensional sphere, to the spectrum of the sublaplacian on M.

CR immersions and Lorentzian geometry

Using tools from Lorentzian geometry (arising from the presence of the Fefferman metric) we prove a Takahashi type theorem (for a class of pseudohermitian immersions covered by connection-preserving equivariant immersions among the total spaces of the canonical circle bundles) thus relating the geometry of a pseudohermitian immersion from a strictly pseudoconvex CR manifold $$M$$ into an odd dimensional sphere, to the spectrum of the sublaplacian on $$M$$ .

CR structures and twisting vacuum spacetimes with two Killing vectors and cosmological constant: type II and more special

Based on the CR formalism of algebraically special spacetimes by Hill, Lewandowski and Nurowski, we derive a nonlinear system of two real ODEs, of which the general solution determines a twisting type II (or more special) vacuum spacetime with two Killing vectors (commuting or not) and at most seven real parameters in addition to the cosmological constant Λ. To demonstrate a broad range of interesting spacetimes that these ODEs can capture, special solutions of various Petrov types are presented and described as they appear in this approach. They include Kerr-NUT, Kerr and Debney/Demiański’s type II, Lun’s types II and III (subclasses of Held–Robinson), MacCallum and Siklos’ type III (Λ < 0) and the type N solutions (Λ ≠ 0) we found in an earlier paper, along with a new class of type II solutions as a nontrivial limit of Kerr and Debney’s type II solutions. Also, we discuss a situation in which the two ODEs can be reduced to one. However, constructing the general solution still remains an open problem.

REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call $$ \mathfrak{n} $$ -reductive. They are orbits of minimal dimension of a compact Lie group K 0 in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration $$ {S^3}\to \mathbb{C}{{\mathbb{P}}^1} $$ . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base.

Reduction of Five-Dimensional Uniformly Levi Degenerate CR Structures to Absolute Parallelisms

Let ${\mathfrak{C}}_{2,1}$ be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. We show that the CR-structures in ${\mathfrak{C}}_{2,1}$ are reducible to ${\mathfrak{so}}(3,2)$ -valued absolute parallelisms and give applications of this fact. Our result is the first instance of reduction to absolute parallelisms for Levi degenerate CR-structures.