Boundary Constructions for CR Manifolds and Fefferman Spaces

The aim of this thesis is to discuss the Cartan boundaries of CR manifolds and their Fefferman spaces.

The Fefferman space of a strictly pseudo-convex CR manifold is defined as the bundle of all real rays in the canonical complex line bundle. Another way of defining it uses the tools of Cartan geometry and leads to a strong relationship between the Cartan geometries of a CR manifold and its Fefferman space. However here the existence of a certain root of the anticanonical complex line bundle is requested which can solely be guaranteed locally. As we are interested in boundaries we need a global construction of Fefferman spaces. Still we find that local results can be transferred from one construction to the other since we have conformal coverings of both.

The Cartan boundary of a manifold is constructed with the help of the corresponding Cartan geometry, which defines a global frame and hence a Riemannian metric on the Cartan bundle which can be completed. Division by the structure group gives the Cartan boundary of the manifold. The Cartan boundary is a generalization of the Cauchy boundary since both coincide in the Riemannian case. In general the Cartan boundary is not necessarily Hausdorff which is not really surprising since boundary phenomena are somehow "singular". For CR manifolds and their Fefferman spaces we especially prove that the projection of the Cartan boundary of the Fefferman space contains the Cartan boundary of the CR manifold.

We finally discuss the Heisenberg group, one of the basic examples of CR manifolds. It is flat but - contrary to the homogeneous space - not compact. We find that the Cartan boundary of the Heisenberg group is a single point and the Cartan boundary of the corresponding Fefferman space is a non degenerate fibre over that point.