Publications

We study CR quadrics satisfying a symmetry property $(\tilde S)$ which is slightly weaker than the symmetry property $(S)$, recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics satisfying the $(\tilde S)$ property in terms of their Levi-Tanaka algebras. In many cases the $(\tilde S)$ property implies the $(S)$ property; this holds in particular for compact quadrics. We also give a new example of a quadric such that the dimension of the algebra of positive-degree infinitesimal automorphisms is larger than the dimension of the quadric.

Let $M$ be a smooth locally embeddable CR manifold, having some CR dimension $m$ and some CR codimension $d$. We find an improved local geometric condition on $M$ which guarantees, at a point $p$ on $M$, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of $p$. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1], Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition, but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension $d > 1$ there are additional phenomena, which are invisible when $d = 1$.

We investigate the CR geometry of the orbits $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a complex flag manifold $X = G/Q$. We are mainly concerned with finite type and holomorphic nondegeneracy conditions, canonical $G_0$-equivariant and Mostow fibrations, and topological properties of the orbits.

We prove a subelliptic estimate Tor systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the authors, and the Hörmander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate.

We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.

We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group $G$ on a flag manifold $G/Q$.

Let $\^G$ be a complex semisimple Lie group, $Q$ a parabolic subgroup and $G$ a real form of $\^G$. The flag manifold $\^G/Q$ decomposes into finitely many $G$-orbits; among them there is exactly one orbit of minimal dimension, which is compact. We study these minimal orbits from the point of view of CR geometry. In particular we characterize those minimal orbits that are of finite type and satisfy various nondegeneracy conditions, compute their fundamental group and describe the space of their global CR functions. Our main tool are parabolic CR algebras, which give an infinitesimal description of the CR structure of minimal orbits.

We prove a relation between the $\bar\partial_M$ cohomology of a minimal orbit $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a flag manifold $G/Q$ and the Dolbeault cohomology of the Matsuki dual open orbit $X$ of the complexification $K$ of a maximal compact subgroup $K_0$ of $G_0$, under the assumption that $M$ is Levi-flat.

In this paper we take up the problem of describing the CR vector bundles $M$ over compact standard CR manifolds $S$, which are themselves standard CR manifolds. They are associated to special graded Abelian extensions of semisimple graded CR algebras.

Let $(V,(.,.))$ be a pseudo-Euclidean vector space and $S$ an irreducible $C\!\ell(V)$-module. An extended translation algebra is a graded Lie algebra $\mathfrak{m}=\mathfrak{m}_{-2}+\mathfrak{m}_{-1}=V+S$ with bracket given by $([s,t],v) = \beta(v\cdot s,t)$ for some nondegenerate $\mathfrak{so}(V)$-invariant reflexive bilinear form $\beta$ on $S$. An extended Poincaré structure on a manifold $M$ is a regular distribution $\mathcal D$ of depth 2 whose Levi form $L_x: \mathcal D_x\wedge\mathcal D_x\rightarrow T_xM/\mathcal D_x$ at any point $x\in M$ is identifiable with the bracket $[.,.]: S\wedge S\rightarrow V$ of a fixed extended translation algebra $\mathfrak m$. The classification of the standard maximally homogeneous manifolds with an extended Poincaré structure is given, in terms of Tanaka prolongations of extended translation algebras and of appropriate gradations of real simple Lie algebras.

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to the case of immersions in $\mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.

We consider a class of compact homogeneous CR manifolds, that we call $\mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3\to\mathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments.

Let $V$ be a complex orthogonal vector space and $S$ an irreducible $C\!\ell(V)$-module. A supertranslation algebra is a Z-graded Lie superalgebra $\mathfrak{m}=\mathfrak{m}_{-2}+\mathfrak{m}_{-1}=V+(S+...+S)$ whose bracket $[.,.]|_{\mathfrak{m}_{-1}\otimes \mathfrak{m}_{-1}}$ is $\mathfrak{so}(V)$-invariant and non-degenerate. We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for $\dim V\geq 3$ and classify them in terms of super-Poincaré algebras and appropriate $\Bbb Z$-gradations of simple Lie superalgebras.

We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds into the Euclidean space $\mathbb{R}^n$ and generalize in a natural way the notion of associated family. We show that the existence of such deformations turns out to be very restrictive and we give a complete classification.

Let $M$ be an orbit of a real semisimple Lie group $G_0$ acting on a complex a flag manifolds $G/Q$ of its complexification $G$. We study the space of global CR functions on $M$ and characterize those $M$ which are strictly locally CR separable, i.e. those for which global CR functions induce local embeddings in $\Bbb C^n$.

We study, from the point of view of CR geometry, the orbits $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a complex flag manifold $G/Q$. In particular we characterize those that are of finite type and satisfy some Levi nondegeneracy conditions. These properties are also graphically described by attaching to them some cross-marked diagrams that generalize those for minimal orbits that we introduced in a previous paper. By constructing canonical fibrations over real flag manifolds, with simply connected complex fibers, we are also able to compute their fundamental group.