Dimensional CR Manifolds Research Articles

Maximal Dimension of Groups of Symmetries of Homogeneous 2-nondegenerate CR Structures of Hypersurface Type with a 1-dimensional Levi Kernel

We prove that for every n ≥ 3 the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, (2n + 1)-dimensional CR manifolds of hypersurface type with a 1-dimensional Levi kernel is equal to n2 + 7, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka’s conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.

Pseudo Cauchy Riemann and Framed Manifolds with Physical Applications

We introduce a pseudo Cauchy Riemann(PCR)-structure defined by a real tensor field $\bar{J}$ of type $(1, 1)$ of a real semi-Riemannian manifold $(\bar{M}, \bar{g})$ such that $\bar{J}^2 = \lambda^2 I$, where $\lambda$ is a function on $\bar{M}$. We prove that, contrary to the even dimensional CR-manifolds, a PCR-manifold is not necessarily of even dimension if $\lambda$ is every where non-zero real function on $\bar{M}$, supported by two odd dimensional examples and one physical model. The metric of PCR-manifold is not severely restricted. Then, we define a pseudo framed(PF)-manifold $(M, g)$ by a real tensor field $f$ such that $f^3 = \lambda^2 f$, where $T(M)$ splits into a direct sum of two subbundles, namely $im(f)$ (with a PCR-structure) and $ ker(f)$, supported by some mathematical and physical examples. Finally, we study a revised version of a contact manifold, called contact PF-manifold, which is a particular case of a PF-manifold where dim$(ker(f))=1$. Contrary to the odd dimensional contact manifolds, there do exist even dimensional contact PF-manifolds. We also propose several open problems.

Self-dual Einstein ACH metrics and CR GJMS operators in dimension three

By refining Matsumoto's construction of Einstein ACH metrics, we construct a one parameter family of ACH metrics which solve the Einstein equation to infinite order and have a given three dimensional CR structure at infinity. When the parameter is 0, the metric is self-dual to infinite order. As an application, we give another proof of the fact that three dimensional CR manifolds admit CR invariant powers of the sublaplacian (CR GJMS operators) of all orders, which has been proved by Gover-Graham. We also prove the convergence of the formal solutions when the CR structure is real analytic.

Diffeomorphisms preservingR-circles in three dimensional CR manifolds

R-circles in (non-degenerate) three dimensional CR manifolds are the analogues to traces of Lagrangian totally geodesic planes on the three sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain Legendrian curves on the manifold. We prove that a diffeomorphism between three dimensional CR manifolds which preserve circles is either a CR diffeomorphism or a conjugate CR diffeomorphism.

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.

Isometric and CR pluriharmonic immersions of three dimensional CR manifolds in Euclidean spaces

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space R n in terms of a Weierstras-like representation. Restricting to the case of immersions in R 4 , we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.

The Space of Cauchy-Riemann Structures on 3-D Compact Contact Manifolds

We study the action of the group of contact diffeomorphisms on CR deformations of compact three dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorphism group in the neighborhood of a fixed embeddable strongly pseudoconvex CR structure.

The Webster scalar curvature problem on the three dimensional CR manifolds

In this paper, we prove some existence results for the Webster scalar curvature problem on the three dimensional CR compact manifolds locally conformally CR equivalent to the unit sphere S 3 of C 2 . Our methods are based on the techniques related to the theory of critical points at infinity.

A Relative Index on the Space of Embeddable CR-Structures, I

We study the problem of embeddability for three dimensional CR-manifolds. Let (M, M0b) denote a compact, embeddable, strictly pseudoconvex CR-manifold and OS the orthogonal projection on ker 0&b. If 16b denotes a deformation of this CR-structure then 1&b is embeddable if and only if

Invariant Kohn-Rossi cohomology and obstruction to embedding of compact real (2n-1)-dimensional CR manifolds in CN

Invariant Kohn-Rossi cohomology and obstruction to embedding of compact real (2n-1)-dimensional CR manifolds in CN

Hölder estimates on domains of complex dimension two and on three dimensional CR manifolds

Hölder estimates on domains of complex dimension two and on three dimensional CR manifolds

A global invariant for three dimensional CR-manifolds

In this note we define a global, R-valued invariant of a compact, strictly pseudoconvex 3-dimensional CR-manifold M whose holomorphic tangent bundle is trivial. The invariant arises as the evaluation of a deRham cohomology class on the fundamental class of the manifold. To construct the relevant form, we start with the CR structure bundle Y over M (see [Ch-Mo], whose notation we follow). The form is a secondary characteristic form of this structure. By fixing a contact form and coframe, i.e., a section of Y, we obtain a form on M. Surprisingly, this form is well-defined up to an exact term, and thus its cohomology class is well-defined in H 3 (M, R). Our motivation for studying this invariant was its analogy with the R/Z secondary characteristic number associated by Chern and Simons to the conforreal class of a Riemannian 3-manifold N, which provides an obstruction to the conformal immersion of N in R 4. Though several formal analogies to the conformal case are valid for our invariant, this one does not hold up: specifically, in w below, we calculate examples which show that the CR invariant can take on any positive real value for hypersurfaces embedded in C 2. It is also clear that the invariant is neither a homotopy nor concordance invariant, but depends in an elusive way on the CR structure. Our inspiration came from the seminal papers of Chern and Moser and Chern and Simons. The idea of looking at secondary characteristic forms of higher order geometric structures in general appears in [Ko-Oc], though with a different intention. In w 2 we will quickly review the definition of a CR structure, the construction of y and its reduction to a pseudo-hermitian structure ~ ld Webster [-We]. In w 3 we define the invariant and prove that it is, in fact, R-valued, and not R/Z-valued as in the Riemannian case. We also prove that if the invariant is stationary as a function of the CR structure, then M is locally CR equivalent to the standard three sphere in C 2, paralleling a result of Chern and Simons. As noted already, w 4 is devoted to the calculation of several examples.

Non-realizable CR structures

The examples given by Nirenberg in [5] and [6] show that not all three dimensional CR manifolds may be realized as real hypersurfaces in 1172. These examples are perturbations of the CR structure on the sphere. In this paper we extend his construction and show that any strictly pseudo-convex realizable CR structure of dimension 3 may be perturbed so as to obtain a non-realizable one. This is done in Sect. 1. In Sect. 2 we show non-realizability for higher dimensional structures provided the Levi form is non-degenerate and has only one positive eigenvalue, In our last section we summarize Cartan's approach to CR geometry. From this viewpoint it was important to decide if Nirenberg's example was really limited to perturbations of the sphere. We have made this paper self-contained and have provided proofs of several known results. For instance Lemma 1.l, and the first part of Lemma 1.5 were proved by Lewy [2]. And Lemma 1.2 is the well known observation that a strictly pseudo-convex hypersurface is convex in some coordinate system,