Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $$G \rtimes S^1$$ G ⋊ S 1 -action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

Regularity of CR mappings of abstract CR structures

We study the [Formula: see text] regularity problem for CR maps from an abstract CR manifold [Formula: see text] into some complex Euclidean space [Formula: see text]. We show that if [Formula: see text] satisfies a certain condition called the microlocal extension property, then any [Formula: see text]-smooth CR map [Formula: see text], for some integer [Formula: see text], which is nowhere [Formula: see text]-smooth on some open subset [Formula: see text] of [Formula: see text], has the following property: for a generic point [Formula: see text] of [Formula: see text], there must exist a formal complex subvariety through [Formula: see text], tangent to [Formula: see text] to infinite order, and depending in a [Formula: see text] and CR manner on [Formula: see text]. As a consequence, we obtain several [Formula: see text] regularity results generalizing earlier ones by Berhanu–Xiao and the authors (in the embedded case).

Cr-Substituted Mesoporous Aluminophosphate Molecular Sieve: Synthesis, Characterization, Crystallization Kinetics and Catalytic Activity

Cr-substituted mesoporous aluminophosphate molecular sieve (Cr-MAP) was synthesized and characterized. Crystallization kinetics curves measured as an index to the relative degree of crystallinity, according to the Arrhenius equation to calculate the apparent nucleation activation energy and crystal growth activation energy of Cr-MAP, which was 63.7 and 14.7 kJ• mol-1, respectively. Cr-MAP had highly catalytic activity for fabricating acetophenone by selectively oxizing ethylbenzene. Using tert-butylhydroperoxide as oxidant and chlorobenzene as solvent at 100 °C for 8 h, acetophenone selectivity, acetophenone yield and ethylbenzene conversion reaches 85.4, 62.2 and 72.8 %, respectively.

Cr-Substituted Mesoporous Aluminophosphate Molecular Sieve: Preparation, Characterization and Catalytic Activity in the Oxidation Reaction of Ethylbenzene

Cr-substituted mesoporous aluminophosphate molecular sieve (Cr-MAP) was prepared and characterized. Cr-MAP is a typical mesoporous molecular sieve with long-range ordered structure, providing effective molecular sieve for fabricating acetophenone by selectively oxizing ethylbenzene with tertiary butyl hydro peroxide (TBHP). When the reaction is at 100 °C for 8 h, using chlorobenzene as solvent and TBHP as oxidant, ethylbenzene conversion, acetophenone selectivity and acetophenone yield reach 72.8 %, 85.4 %, and 62.2 %, respectively.

MINIMALITY, HARMONICITY AND CR GEOMETRY FOR REEB VECTOR FIELDS

Let (M, g) be a Riemannian manifold and T1 M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T1M the Sasaki metric [Formula: see text]. This metric, and other well-known Riemannian metrics on T1 M, are particular examples of Riemannian natural metrics. In this paper we equip T1 M with a Riemannian natural metric [Formula: see text] and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kähler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any [Formula: see text] that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map [Formula: see text] for any Riemannian natural metric [Formula: see text]. Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then [Formula: see text] is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and [Formula: see text] is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T1S2n+1 depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S2n+1 to T1S2n+1.