Embeddability of smooth Cauchy-Riemann manifolds

Summary The main purpose of this paper is to give a sufficient condition for global embeddability of smooth Cauchy-Riemann manifolds (CR-manifolds) into complex manifolds with boundary. Namely, let M be a smooth CR-manifold of real dimension 2n – 1 and CR-dimension n – 1, where n 2, which is locally CR-embeddable into a complex manifold. Assume further that the Levi form of M is non-vanishing at each point. The main result of this paper is that such a CR-manifold is globally CR-embeddable into an n-dimensional complex manifold with boundary. Moreover if the Levi form has at each point of M eigenvalues of opposite signs, then M embeds into a complex manifold without boundary.This research is supported by a grant from Consiglio Nazionale delle Ricerche in Italy.     

Pompeiu transforms on geodesic spheres in real analytic manifolds

We prove a support theorem for Pompeiu transforms integrating on geodesic spheres of fixed radiusr>0 on real analytic manifolds when the measures are real analytic and nowhere zero. To avoid pathologies, we assume thatr is less than the injectivity radius at the center of each sphere being integrated over. The proof of the main result is local and it involves the microlocal properties of the Pompeiu transform and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities.     

Regularity of\bar \partial _b , complex on nondegenerate Cauchy-Riemann manifolds, complex on nondegenerate Cauchy-Riemann manifolds

For each point ξ in a CR manifold M of codimension greater than 1, the CR structure of M can be approximated by the CR structure of a nilpotent Lie group G_ξ of step two near ξ. G_ξ varies with ξ. $\square _b $ and $\bar \partial _b $ on M can be approximated by $\square _b $ and $\bar \partial _b $ on the nilpotent Lie group G_ξ We can construct the parametrix of $\square _b $ on M by using the parametrix of $\square _b $ on nilpotent group of step two, and define a quasidistance on M by the approximation. The regularity of $\square _b $ and $\bar \partial _b $ follows from the Harmonic analysis on M.     

Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds

We study a class of submanifolds, called Generalized Cauchy-Riemann (GCR) lightlike submanifolds of indefinite Sasakian manifolds as an umbrella of invariant, screen real, contact CR lightlike subcases [8] and real hypersurfaces [9]. We prove existence and non-existence theorems and a characterization theorem on minimal GCR-lightlike submanifolds.     

Local Cauchy-Riemann embeddability of real hyperboloids into spheres

In this paper, we study the local Cauchy-Riemann embeddability of strictly pseudoconvex real hyperboloids into spheres. By solving a CR analogue of the Gauss equation, we prove that is CR-embeddable into spheres with a CR co-dimension if and only if it is spherical.

     

A note on the instability of embeddings of Cauchy-Riemann manifolds

We prove that there are compact strictly pseudoconvex CR manifolds, embedded into some Euclidean space, that admit small deformations that are also embeddable but their embeddings cannot be chosen close to the original embedding. Both authors were partially supported by NSF grants.     

On the dynamics of certain actions of free groups on closed real analytic manifolds

Let M be a closed connected real analytic manifold; let be a free group on two generators. The set of analytic actions of on M endowed with Taken‚s topology contains a nonempty open subset whose corresponding actions share three properties: (a) they have every orbit dense, (b) they leave invariant no geometric structure on M, (c) any homeomorphism conjugating two of them is analytic. Received: October 3, 2001     

All compact manifolds are homeomorphic to totally algebraic real algebraic sets

Both authors supported in part by the N.S.F.     

Diabatic limit, eta invariants and Cauchy-Riemann manifolds of dimension 3

We relate a recently introduced non-local invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various η-invariants: on the one hand a renormalized η-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the η-invariant of the middle degree operator of the contact complex. We then provide explicit computations for transverse circle invariant CR structures on Seifert manifolds. This yields obstructions to filling a CR manifold by complex hyperbolic, Kähler-Einstein, or Einstein manifolds.     

Cellular homology of real flag manifolds

Let ${\mathbb{F}}_{\Theta }=G∕P_{\Theta }$ be a generalized flag manifold, where $G$ is a real non-compact semi-simple Lie group and $P_{\Theta }$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow ${\mathbb{F}}_{\Theta }$ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in ${\mathbb{R}}^n$ and use them to compute the boundary operator $\partial$ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of $\partial$ are 0 or $\pm 2$ (so that ${\mathbb{Z}}_2$-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.     

Toda flows and real Hessenberg manifolds

We relate the Toda flow on the “p-part” of a semi-simple Lie algebra to the topology of real Hessenberg manifolds, and we obtain their mod2 Betti numbers by reversing Morse inequalities using a theorem of Floyd and a result on the Weyl group.