Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds

 In this paper we study warped product CR-submanifolds in Kaehler manifolds and introduce the notion of CR-warped products. We prove several fundamental properties of CR-warped products in Kaehler manifolds and establish a general inequality for an arbitrary CR-warped product in an arbitrary Kaehler manifold. We then investigate CR-warped products in a general Kaehler manifold which satisfy the equality case of the inequality. Finally we classify CR-warped products in complex Euclidean space which satisfy the equality. (Received 24 August 2000; in revised form 19 February 2001)     

On a class of symmetric CR manifolds

We study and classify a large class of minimal orbits in complex flag manifolds for the holomorphic action of a real Lie group. These orbits are all symmetric CR spaces for the restriction of a suitable class of Hermitian invariant metrics on the ambient flag manifold. As a particular case we obtain that the standard compact homogeneous CR manifolds associated with semisimple Levi-Tanaka algebras are symmetric CR-spaces.     

Normal <Emphasis Type="Italic">CR</Emphasis> structures on S

We classify the normal CR structures on S³ and their automorphism groups. Together with [2], this closes the classification of normal CR structures on compact 3-manifolds. We give a criterion to compare two normal CR structures, and we show that the underlying contact structure is, up to diffeomorphism, unique. Mathematics Subject Classification (2000): 53A40, 53C25, 53D10. Received: 9 April 2002; in final form: 8 May 2002 // Published online: 20 March 2003     

Contact CR-Warped Product Submanifolds in Sasakian Space Forms

Recently, B.-Y. Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. Afterwards, I. Hasegawa and the present author obtained a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Sasakian manifolds. In this paper, we improve the above inequality for contact CR-warped products in Sasakian space forms. Some applications are derived. A classification of contact CR-warped products in spheres, which satisfy the equality case, identically, is given.Mathematics Subject Classifications (2000). 53C40, 53C25.     

Complete nondegenerate locally standard CR manifolds

Abstract. Let M be a complete nondegenerate locally standard CR manifold. We show that a necessary and sufficient condition for M to be compact is that the Lie algebra of its infinitesimal CR automorphisms is semisimple. In general we realize M as a Mostow fibration over a compact CR manifoldB whose universal covering is a Cartesian product of Hermitian symmetric spaces and compact nondegenerate standard CR manifolds. Received: 22 July 1998 / Published online: 8 May 2000     

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday     

On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001     

Asymptotical flatness and cone structure at infinity

We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends, and we classify (except for the case dim M=4, where it remains open if one of the theoretically possible cones can actually arise) for simply connected ends all possible cones at infinity. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat m-manifold with nonnegative sectional curvature is isometric to , whereS is an asymptotically flat surface. Received: 5 January 2000 / Published online: 19 October 2001     

Global Homotopy Formulas on <Emphasis Type="Italic">q</Emphasis>-concave <Emphasis Type="Italic">CR</Emphasis> Manifolds for Small Degrees

Using functional analysis, we derive from local homotopy formulas for the tangential Cauchy-Riemann operator a global homotopy formula for compact CR manifolds without loss of regularity.      

An Andreotti–Vesentini separation theorem on real hypersurfaces

Using Serre duality in CR manifolds and integral operators for the solution of the tangential Cauchy–Riemann equation with compact support, we prove a separation theorem of Andreotti–Vesentini type for the -cohomology in q-concave real hypersurfaces. Received: February 17, 1999?Published online: May 10, 2001     

Mok-Siu-Yeung type formulas on contact locally sub-symmetric spaces

We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps.      

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

 A CR-submanifold N of a Kaehler manifold is called a CR-warped product if N is the warped product of a holomorphic submanifold and a totally real submanifold of . This notion of CR-warped products was introduced in part I of this series. It was proved in part I that every CR-warped product in a Kaehler manifold satisfies a basic inequality: . The classification of CR-warped products in complex Euclidean space satisfying the equality case of the inequality is archived in part I. The main purpose of this second part of this series is to classify CR-warped products in complex projective and complex hyperbolic spaces which satisfy the equality.     

Doubly warped product CR -submanifolds in locally conformal K?hler manifolds

In this paper we study doubly warped product CR submanifolds in locally conformal K?hler manifolds, and we found a B.Y. Chen’s type inequality for the second fundamental form of these submanifolds.     

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

 A CR-submanifold N of a Kaehler manifold is called a CR-warped product if N is the warped product of a holomorphic submanifold and a totally real submanifold of . This notion of CR-warped products was introduced in part I of this series. It was proved in part I that every CR-warped product in a Kaehler manifold satisfies a basic inequality: . The classification of CR-warped products in complex Euclidean space satisfying the equality case of the inequality is archived in part I. The main purpose of this second part of this series is to classify CR-warped products in complex projective and complex hyperbolic spaces which satisfy the equality. (Received 13 March 2001; in revised form 10 August 2001)     

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.     

Submanifolds in differentiable manifolds endowed with differential-geometric structures VI.CR-submanifolds in a manifold of almost-complex structure

This survey is an exposition of the results of the study ofCR-submanifolds in almost-Hermitian manifolds and their subclasses. The concept of aCR-submanifold is generalized to the case of almost-complex manifolds without a metric.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 19, pp. 59–100, 1987.     

Local exactness of top-degree forms in abstract Cauchy-Riemann complexes

 We show that the Poincaré lemma for the tangential Cauchy-Riemann operator in the top degree is valid in abstract 1-concave CR manifolds. Received: 1 July 2002 Published online: 24 January 2003     

Doubly warped product <Emphasis Type="Italic">CR</Emphasis>-submanifolds in locally conformal Kähler manifolds

In this paper we study doubly warped product CR submanifolds in locally conformal K?hler manifolds, and we found a B.Y. Chen’s type inequality for the second fundamental form of these submanifolds. Beneficiary of a CNR-NATO Advanced Research Fellowship pos. 216.2167 Prot. n. 0015506.     

A partially penalized P1/CR immersed finite element method for planar elasticity interface problems

We propose a partially penalized P₁/CR immersed finite element (IFE) method with midpoint values on edges as degrees of freedom for CR elements to solve planar elasticity interface problems. Optimal approximation errors in L² norm and H¹ semi‐norm are obtained for the P₁/CR IFE spaces. Moreover, by adding some stabilization terms on the edges of interface elements, we derive an optimal error estimate for the P₁/CR IFE method. Our method differs from the method with average values on edges as degrees of freedom for P₁/CR elements in Qin et al.'s study, where no approximation theoretical result was presented. Numerical examples confirm our theoretical results.     

Maximal L 2 and Pointwise Hölder Estimates for □ b on CR Manifolds of Class C 2

In this paper we prove the optimal pointwise Hölder estimates for the Kohn's Laplacian □ _b for CR manifolds of class C ². We first derive optimal L ² theory for solutions of □ _b directly without using pseudo-differential operators. Then we use the scaling method to obtain pointwise estimates in non-istotropic Campanato spaces based on the maximal L ² theory.