arXiv research

Class I CR manifolds have initial G-structure a certain 4-dimensional subgroup of GL_3(C). Class II CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_4(C). Class III-1 CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_5(C). Class III-2 CR manifolds have initial G-…

Study proves curvature flow existence on CR manifolds.

Study on spherical CR manifolds with non-trivial Chern classes.

Characterizes CR manifolds as critical points of an energy functional.

Study CR Yamabe constant, flow, and soliton on CR manifolds.

Paper proves nonnegativity of CR Paneitz operator for embeddable CR manifolds.

Study Szegő kernel on non-compact CR manifolds with specific conditions.

New eigenvalue estimate for CR manifolds' Kohn-Dirac operator.

Characterizes CR manifolds in complex flag manifolds.

We prove several theorems concerning the connection between the local CR embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dime…

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

The paper studies a flow to prescribe curvature on CR manifolds.

We classify all compact simply connected homogeneous CR manifolds $M$ of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group $G(M)$ of automorphisms of $M$. We characterize also the standa…

New CR manifolds found with same Kohn Laplacian spectra.

The Fefferman metric connects CR manifolds to conformal geodesics in 3D.

Study vanishing theorems for CR manifolds using contact forms and Laplacian formulas.

Analyzes the Levi form on CR manifolds of any dimension.

We show that various notions of local homogeneity for CR-manifolds are equivalent. In particular, if germs at any two points of a CR-manifold are CR-equivalent, there exists a transitive local Lie group action by CR-automorphisms near every point.

Study on CR Paneitz operator on non-embeddable CR manifolds.

Study on spectral asymptotics of Toeplitz operators on CR manifolds.

We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\leq p\leq q$, 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 $(1,n-1)$ which is not locally CR flat and admits …

The paper studies heat kernel asymptotics for Kohn Laplacians on CR manifolds.

Researchers create normal forms for CR manifolds in complex space.

Embeds CR manifolds into complex spaces using equivariant actions.

We study the equivalence problem for $4$-dimensional CR-manifolds of CR-dimension $1$ and codimension $2$ which are referred to as Engel CR-manifolds. We construct a canonical Cartan connection on such CR-manifolds through Cartan equivalence's method. In particular, we give the explicit expression of $4$ biholomorphic …

We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR manifold.

For compact CR manifolds of hypersurface type which embed in complex projective space, we show that for all k large enough there exist linear systems of ${\mathcal{O}}(k)$ which when restricted to the CR manifold are generic in a suitable sense. These systems are constructed using approximately holomorphic geometry.

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equivariant K…

Study on higher order Levi forms on homogeneous CR manifolds, improving previous results.

We study the pseudohermitian sectional curvature of a CR manifold.

Vanishing theorem on CR manifolds with non-negative curvature.

Formula for Toeplitz operator kernel on CR manifolds.

The purpose of this work is to propose a mixed Hodge structure over a CR manifold. As you know, for a CR manifold, Kohn-Rossi cohomology is naturally introduced. However, the relation between Kohn-Rossi cohomology and De Rham cohomology is not so well understood, even in Tanaka's work. We discuss this point.

Geometric quantization studied on CR manifolds with Lie group actions.

In this paper we study the topology of pseudo convex CR manifolds whose Reeb flow preserves the Levi metric.

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.

We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded En…

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core …

We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solv…

New CR-structures lemma simplifies CR-manifold deformation proof.

We obtain a Bochner type formula and an estimate from below on the spectrum of the sublaplacian of a compact strictly pseudoconvex CR manifold.

A method is proposed to obtain examples of smooth CR-manifolds whose local stability group is neither a Lie group nor infinite-dimensional.

Study on contact forms with constant curvature on CR manifolds.

Researchers study surface area functionals in CR manifolds, deducing equations for various cases.

R-circles in general three dimensional CR manifolds (of contact type) are the analogues to traces of Lagrangian totally geodesic planes on the 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 betwee…

Let M be a CR manifold of hypersurface type, which is Levi degenerate but also satisfying a k-nondegeneracy condition at all points. This might be only if dim M is greater than or equal to 5 and if dim M = 5, then k= 2 at all points. We prove that for any 5-dimensional, uniformly 2-nondegenerate CR manifold M there exi…

The CR Frankel conjecture is proven for spherical CR manifolds.

Let $X$ be an orientable compact Levi-flat CR manifold and let $L$ be a positive CR complex line bundle over $X$. We prove that certain microlocal conjugations of the associated Szegő kernel admits an asymptotic expansion with respect to high powers of $L$. As an application, we give a Szegő kernel proof of the Kodaira…