At the boundary of asymptotically complex hyperbolic manifolds

The complex hyperbolic space is the complex counterpart of real hyperbolic geometry, and is the simplest instance of a negatively curved Kähler-Einstein manifold. Similarly to the real case, the Riemannian geometry of the complex hyperbolic space is in one-to-one correspondance with the conformal geometry of its boundary at infinity, which is a strictly pseudoconvex Cauchy-Riemann (CR) structure. This correspondance has proven useful to the study of complex domains as well as that of Kähler manifolds.

In this talk, we consider a complete, non-compact almost Hermitian manifold whose geometry at infinity is locally modelled on that of the complex hyperbolic space. Under purely geometric considerations, we will prove that such a manifold admits a natural compactification at infinity by a strictly pseudoconvex CR structure.