CR Manifolds Research Articles

On submersions of the complex indicatrix

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

Ruled three-dimensional CR submanifolds of the sphere S6(1)

We investigate proper, three-dimensional CR submanifolds of the nearly Kahler sphere S6(1) ruled by totally geodesic spheres S2(1), and classify them by using a sphere curve and a vector field along that curve.

$Q$-prime curvature and scattering theory on strictly pseudoconvex domains

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. In this paper, we generalize these objects to the boundaries of asymptotically complex hyperbolic Einstein (ACHE) manifolds, which are partially integrable, strictly pseudoconvex CR manifolds, by using the scattering matrix for ACHE manifolds. In this setting, the $P$-prime operator is a self-adjoint pseudodifferential operator acting on smooth functions and the $Q$-prime curvature is globally determined by the ACHE manifold and the choice of a contact form on the boundary. We prove that the integral of the $Q$-prime curvature is a conformal primitive of the $Q$-curvature; in particular, it defines an invariant of ACHE manifolds whose boundaries admit a contact form with zero $Q$-curvature. We also apply the generalized $Q$-prime curvature to compute the renormalized volume of strictly pseudoconvex domains whose boundaries may not admit pseudo-Einstein structure.

Szegő kernel asymptotics and Kodaira embedding theorems of Levi-flat 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 Szeg\H{o} kernel admits an asymptotic expansion with respect to high powers of $L$. As an application, we give a Szeg\H{o} kernel proof of the Kodaira type embedding theorem on Levi-flat CR manifolds due to Ohsawa and Sibony.

A CONCRETE APPROACH TO DIAGONAL SHORT TIME ASYMPTOTICS OF HEAT KERNELS ASSOCIATED WITH SUB-LAPLACIAN ON CR MANIFOLDS

A diffusion process associated with the real sub-Laplacian $\Delta_b$, the real part of the complex Kohn-Spencer Laplacian $\square_b$, on a strictly pseudoconvex CR manifold has been constructed. In this paper, we investigate diagonal short time asymptotics of the heat kernel corresponding to the diffusion process by using Watanabe's asymptotic expansion and give a representation for the asymptotic expansion of heat kernels which shows a relationship to the geometric structure.

Inflexible CR submanifolds

In this paper we introduce the concept of inflexibleCR submanifolds. These are CR submanifolds of some complex Euclidean space such that any compactly supported CR deformation is again globally CR embeddable into some complex Euclidean space. Our main result is that any 2-pseudoconcave quadratic CR submanifold of type (n, d) in \(\mathbb {C}^{n+d}\) is inflexible.

Regularity of infinitesimal CR automorphisms

We study the regularity of infinitesimal CR automorphisms of abstract CR structures which possess a certain microlocal extension and show that there are smooth multipliers, completely determined by the CR structure, such that if [Formula: see text] is such an infinitesimal CR automorphism, then [Formula: see text] is smooth for all multipliers [Formula: see text]. As an application, we study the regularity of infinitesimal automorphisms of certain infinite type hypersurfaces in [Formula: see text].

Variation of total Q-prime curvature on CR manifolds

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Kähler–Einstein metric.

The Yamabe problem on noncompact CR manifolds

The Yamabe problem on noncompact CR manifolds

Distributional boundary values of holomorphic functions on product domains

We show that holomorphic functions of polynomial growth on domains with corners have distributional boundary values in an appropriate sense, provided the corners are generic CR manifolds. We also prove an analog of the Bochner–Hartogs theorem for these boundary values for the simplest such domains, the product domains.

Linearized pseudo-Einstein equations on the Heisenberg group

We study the pseudo-Einstein equation R 1 1 ¯ = 0 on the Heisenberg group H 1 = C × R . We consider first order perturbations θ ϵ = θ 0 + ϵ θ and linearize the pseudo-Einstein equation about θ 0 (the canonical Tanaka–Webster flat contact form on H 1 thought of as a strictly pseudoconvex CR manifold). If θ = e 2 u θ 0 the linearized pseudo-Einstein equation is Δ b u − 4 | L u | 2 = 0 where Δ b is the sublaplacian of ( H 1 , θ 0 ) and L ¯ is the Lewy operator. We solve the linearized pseudo-Einstein equation on a bounded domain Ω ⊂ H 1 by applying subelliptic theory i.e. existence and regularity results for weak subelliptic harmonic maps. We determine a solution u to the linearized pseudo-Einstein equation, possessing Heisenberg spherical symmetry, and such that u ( x ) → − ∞ as | x | → + ∞ .

The second CR Yamabe invariant

Let (M,θ) be a closed, connected, strictly pseudoconvex CR manifold with dimension 2n+1≥3. We define the second CR Yamabe invariant in terms of the second eigenvalue of the Yamabe operator and the volume of M over the pseudo-convex pseudo-hermitian structures θ conformal to θ. Then we study when it is attained and classify the CR-sphere by its second CR Yamabe invariant. This work is motivated by the work of B. Ammann and E. Humbert [1] on the Riemannian context.

Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

Let \(M\subset {\mathbb {C}}^N\) and \(M'\subset {\mathbb {C}}^{N'}\) be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also essentially necessary, for all sufficiently smooth CR maps \(h:U\rightarrow M'\) defined on a connected open subset of M and of rank larger than a prescribed integer r to be real-analytic on a dense open subset of U. This condition corresponds to the nonexistence of nontrivial holomorphic deformations of germs of real-analytic CR mappings whose rank is larger than r. As a consequence, we obtain several new results about analyticity of CR mappings that, at the same time, generalize and unify a number of previous existing ones.

Almost CR quaternionic manifolds and their immersibility in $$\mathbb {H}$$ H P $$^n$$ n .

We apply the general theory of codimension one integrability conditions for G-structures developed in Santi (Ann Mat Pura Appl 195:1463–1489, 2016) to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold.

Four-dimensional CR submanifolds of the sphere S6(1) with two-dimensional nullity distribution

We investigate four-dimensional CR submanifolds of the nearly Kähler sphere S6(1) with nullity distribution of the maximal possible dimension two, and classify them using a sphere curve and a vector field along that curve.

A class of four dimensional CR submanifolds of the sphere S6(1)

In this paper, we investigate four dimensional CR submanifolds of the nearly Kähler sphere S 6 ( 1 ) that locally have a twisted product structure of special type with respect to their CR structure, and explicitly construct them by using a sphere curve. We show that four dimensional CR submanifolds with two dimensional nullity distribution belong to this class if their totally real distribution is totally geodesic and describe them.

The CR-geometry of the complex indicatrix

The CR-geometry of the complex indicatrix

Umbilical points on three dimensional strictly pseudoconvex CR manifolds I: manifolds with U(1)-action

The question of existence of umbilical points, in the CR sense, on compact, three dimensional, strictly pseudoconvex CR manifolds was raised in the seminal paper by S.-S. Chern and J. K. Moser in 1974. In the present paper, we consider compact, three dimensional, strictly pseudoconvex CR manifolds that possess a free, transverse action by the circle group $U(1)$. We show that every such CR manifold $M$ has at least one orbit of umbilical points, {\it provided} that the Riemann surface $X:=M/U(1)$ is not a torus. In particular, every compact, circular and strictly pseudoconvex hypersurface in $\mathbb C^2$ has at least one circle of umbilical points. The existence of umbilical points in the case where $X$ is a torus is left open in general, but it is shown that if such an $M$ has additional symmetries, in a certain sense, then it must possess umbilical points as well.

A Promising Catalytic and Theranostic Agent Obtained through the In‐Situ Synthesis of Au Nanoparticles with a Reduced Polyoxometalate Incorporated within Mesoporous MIL‐101

A well‐controlled three‐step green synthetic method allows the synthesis of composite materials based on the highly stable chromium(III) terephthalate MIL‐101(Cr) with gold nanoparticles (NPs) and polyoxometalates (POMs) inside its mesopores. First, the POM was included within the MIL‐101(Cr) cavities through the direct synthesis of MIL‐101(Cr) in the presence of the preformed H3PMo12O40 POM. Then, the incorporated POM was reduced in situ with H2. Finally, an in‐situ soft reduction of HAuCl4 by the reduced POM led to the formation of Au NPs and oxidized POM within the MIL‐101(Cr) structure [(Au)H2redPOM@101]. A final Au/POM atomic ratio of 1.8 was reached. The chemical and structural stabilities of both MIL‐101(Cr) and the POM were preserved during the whole process. In addition to the potential catalytic properties of these highly porous composite systems, we have demonstrated their promising performance as optical contrast agents. The (Au)H2redPOM@101 composites exhibited excellent biostability and were rapidly internalized in macrophage cells, as observed through fluorescence confocal microscopy.

Global Real Analyticity of the Kohn–Laplacian on Pseudoconvex CR Manifolds with Comparable Levi Form

In this paper, we study the global analytic hypoellipticity of the \(\Box _b\) operator on a general CR manifold of real dimension \((2n-1)\), with \(n\ge 3\). In particular, if M is a compact, real analytic, pseudoconvex CR manifold satisfying the \(D^{\epsilon }(q)\) and \((CR-P_q)\) conditions, and a special property for vector field T, we consider the following nonelliptic partial differential equation $$\begin{aligned} \Box _bu=f. \end{aligned}$$ If f is globally real analytic, we conclude that u is globally real analytic as well. The methods applied in this work are inspired from Tartakoff (1(4):283–311, 1976) and Tartakoff (89–90:85–116 1981).