CR Dimension Research Articles

Quasi Yamabe Solitons on CR Submanifolds of Maximal CR Dimension in Kähler Manifolds

In this paper we give necessary and sufficient conditions for a CR submanifold of maximal CR dimension in arbitrary Kähler manifold to admit (quasi-)Yamabe structure, with naturally chosen soliton vector field. When the ambient manifold is a non-flat complex space form, we give a complete classification of such solitons, under certain conditions.

THE CURRENT AND RETROSPECTIVE COGNITIVE RESERVE (2CR) SURVEY AND ITS RELATIONSHIP WITH COGNITIVE AND MOOD MEASURES

Abstract Cognitive reserve (CR) is often assessed with surveys spanning demographic, lifestyle, and socio-behavioral variables. The role of both past and current life experiences on CR has, however, rarely been examined. We developed the Current and Retrospective Cognitive Reserve (2CR) survey to assess classical CR proxies (socio-economic status, engagement in leisure and social activities) and other dimensions of potential importance (family engagement, religious/spiritual activity) both currently (CRc; in later adulthood) and retrospectively (CRr; as recalled from younger adulthood). We administered the 2CR, measures of general cognitive functioning, working memory (WM), vocabulary, fluid reasoning, and depressive symptoms (DS) to 235 community-dwelling Italian adults (ages 55-90 years). We used exploratory and confirmatory factor analyses to examine the 2CR latent structure, and we estimated correlations of its dimensions with cognitive abilities and DS. Analyses confirmed a three-level factor structure with two global CR factors (CRc and CRr) at the top level, dimensional CR factors (socio-economic status, family engagement, leisure activity, social engagement, and religious/spiritual activity) at mid-level, and observed items at the lowest level. Item-factor representations partially differed across CRc and CRr. Both CRc and CRr were positively correlated with measures of intelligence, WM, and DS. Associations with measures of intelligence were comparatively stronger for CRr, whereas associations of WM and DS were slightly stronger for CRc. The 2CR can be considered a reliable survey for assessing CR proxies within a multidimensional, “life stage-dependent” framework insofar as CRc are CRr closely related but also differently associated with intelligence, WM, and DS.

C-R Immersions and Sub-Riemannian Geometry

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm of the Reeb vector field T of (M, θ) is of order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction gϵA of the Levi form GΘ, we write the embedding equations for the immersion f:M→A,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ→0+. For every isopseudohermitian immersion f:M→S2N+1 into a sphere S2N+1⊂CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R≤2n(f∗gΘ)(T,T)+n+1+12{∥H(f)∥gΘf2+∥traceGθΠH(M)∇⊤−∇∥f∗gΘ2} with equality if and only if B(f)=0 and ∇⊤=∇ on H(M)⊗H(M). This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.

The scalar curvature of CR submanifold of maximal CR dimension with a conformal change

In this paper, we investigate the codimension reduction theorem for an n-dimensional submanifold of an (n + p)-dimensional manifold (M, ~g) with a conformal change g = exp(–f)g where g denotes the Fubini-study metric on (n + p)-dimensional complex projective space P n+p/2 (C). Moreover, we tend to calculate the scalar curvature of an n-dimensional CR submanifold of maximal CR dimension of (M; g) and achieve the sufficient conditions for the existence of a totally geodesic submanifold M0 that includes M.

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.

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.

Functional determinant on pseudo-Einstein 3-manifolds

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,\theta)$, we establish an expression for the difference of determinants of the Paneitz type operators $A_{\theta}$, related to the problem of prescribing the $Q'$-curvature, under the conformal change $\theta\mapsto e^{w}\theta$ with $w\in \P$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four dimensional Riemannian manifolds established in \cite{BO2}. We also provide an existence result of maximizers for the scaling invariant functional determinant as in \cite{CY}.

Renormalized characteristic forms of the Cheng–Yau metric and global CR invariants

For each invariant polynomial Φ, we construct a global CR invariant via the renormalized characteristic form of the Cheng–Yau metric on a strictly pseudoconvex domain. When the degree of Φ is 0, the invariant agrees with the total Q′-curvature. When the degree is equal to the CR dimension, we construct a primed pseudo-hermitian invariant IΦ′ which integrates to the corresponding CR invariant. These are generalizations of the I′-curvature on CR five-manifolds, introduced by Case–Gover.

Certain submanifolds of complex space forms

After recalling the geometric meaning of the commutativity of the second fundamental tensor of a real hypersurface of a complex space form and its induced almost contact structure, we present some classification theorems for CR submanifolds of maximal CR dimension and submanifolds of real codimension two of complex space forms $${\overline{M}}$$, under the algebraic condition on the second fundamental form of the submanifold and the endomorphism induced from the natural almost complex structure of $${\overline{M}}$$ on the tangent bundle of the submanifold.

Contact CR Submanifolds of maximal Contact CR dimension of Sasakian Space Form

In this paper, we investigate contact CR submanifolds of contact CR dimension in Sasakian space form and introduce the general structure of these submanifolds and then studying structures of this submanifols with the condition  h(FX,Y)+h(X,FY)=g(FX,Y)zeta, for the normal vector field zeta, which is nonzero, and we classify these submanifolds../files/site1/files/61/0Abstract.pdf

A Short Proof of the Maximum Conjecture in CR Dimension one

Introduction In this paper and by employing some certain techniques and results arisen in the theory of Tanaka, we provide a short proof for the maximum conjecture on the rigidity of Beloshapka's models of the specific CR dimension one and of length . As a consequence, we realize that the Lie group of biholomorphisms (CR automorphisms) associated with each of these models only consists of linear maps. Material and methods Two major approaches employed so far to study Beloshapka's maximum conjecture have been 1) the approach of envelope of holomorphy, applied by Ilya Kossovskii to confirm the mentioned conjecture in the specific lengths three and four and 2) Cartan's approach of solving (biholomorphic) equivalence problems, employed by the present author in CR dimension one. Both of these approaches are geometric. In this paper, we introduce and employ the algebraic approach of applying some techniques in Tanaka's theory of transitive prolongations to study the already mentioned conjecture in the specific CR dimension one. The proofs are based upon some results concerning Tanaka prolongation of rank two fundamental algebras of lengths , achieved by Medori and Nacinovich (1997). Here, we prove first an algebraic parallel version of Beloshapka's conjecture and employ the results to solve this geometric open problem in CR dimension one. Results and discussion As is the main goal of this paper, we confirm the maximum conjecture on the rigidity of Beloshapka's CR models in CR dimension one. As a consequence, we realize that each biholomorphic deformation of these models is linear. It may be worth to notice that the maximum conjecture in CR dimension one has been confirmed before by the present author, using the Cartan approach of solving equivalence problems. But, here we provide a much shorter proof for this result. Conclusion The following conclusions were drawn from this research.

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.

Totally Nondegenerate Models and Standard Manifolds in CR Dimension One

It is shown that two Levi–Tanaka and infinitesimal CR automorphism algebras, associated with a totally nondegenerate model of CR dimension one are isomorphic. As a result, the model surfaces are maximally homogeneous and standard. This gives an affirmative answer in CR dimension one to a certain question formulated by Beloshapka.

Aspects of the Levi form

We discuss various analytical and geometrical aspects of the Levi form, which is associated with a CR manifold having any CR dimension and any CR codimension.

Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds

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 solve $$\square _b$$ works whenever $${\overline{\partial }}_b$$ on the compact CR manifold has closed range in $$L^2$$ ; in particular, as in Beals and Greiner (Calculus on Heisenberg manifolds. Annals of mathematics studies, vol 119. Princeton University Press, Princeton, 1988), it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in $${\mathbb {C}}^2$$ . Our result provides in turn a key step in the proof of a positive mass theorem in 3-dimensional CR geometry, by Cheng et al. (Adv Math 308:276–347, 2017), which they then applied to study the CR Yamabe problem in 3 dimensions.

An integral formula for the $Q$-prime curvature in 3-dimensional CR geometry

We give an integral formula for the total $Q^\prime$-curvature of a three-dimensional CR manifold with positive CR Yamabe constant and nonnegative Paneitz operator. Our derivation includes a relationship between the Green's functions of the CR Laplacian and the $P^\prime$-operator.

Biholomorphic equivalence to totally nondegenerate model CR manifolds

Applying Elie Cartan’s classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka’s model of CR dimension one and codimension $$k> 1$$ , whence of real dimension $$2+k$$ , is reducible to some absolute parallelism, namely to an $$\{e\}$$ -structure on a certain prolonged manifold of real dimension either $$3+k$$ or $$4+k$$ . The proof relies upon a careful weight analysis on the structure equations associated with the mentioned problem of equivalence. As one of the applications of the achieved results, we also reconfirm in CR dimension one Beloshapka’s maximum conjecture on the holomorphic rigidity of his models in certain lengths equal or greater than three.

Estimates for the complex Green operator: Symmetry, percolation, and interpolation

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees, that is, they hold for $(p,q)$--forms if and only if they hold for $(m-p,m-1-q)$--forms. Here $m$ equals the CR dimension of $M$ plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex, i.e. if they hold for $(p,q)$--forms, they also hold for $(p,q+1)$--forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on $(p,q_{1})$--forms and on $(p,q_{2})$--forms ($q_{1}\leq q_{2}$), then it is compact on $(p,q)$--forms for $q_{1}\leq q\leq q_{2}$. It is interesting to contrast this behavior of the complex Green operator with that of the $\overline{\partial}$--Neumann operator on a pseudoconvex domain.

On the maximum conjecture

Abstract We verify the maximum conjecture on the rigidity of totally nondegenerate model CR manifolds in the following two cases: (i) for all models of CR dimension one, (ii) for the so-called full-models, namely those in which their associated symbol algebras are free CR. In particular, we discover that in each arbitrary CR dimension and length ≥ 3 {\geq 3} , there exists at least one totally nondegenerate model, enjoying this conjecture. Our proofs rely upon some recent results in the Tanaka theory of transitive prolongation of fundamental algebras.

Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler–Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization.