We give a homotopy formula for the $\partial {̅}_b$ operator on a domain with (q+k)-concave boundary. As a consequence we show that the dimension of the $\partial {̅}_b$-cohomology groups for some CR manifolds q-concave at infinity is finite.

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n 2 are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n 2)-dimensional submanifolds in ${ℂ}^{n+n^2}$ for all n > 1. In...