Alan Pinoy, KTH Royal Institute of Technology
Conformally compact manifolds form a natural setting for the study of asymptotically hyperbolic metrics, and in particular Poincaré-Einstein metrics. This setting has proven to be fruitful in order to construct conformal invariants. The boundary at infinity is however often given as an had oc data. During the 2010’s, Bahuaud-Gicquaud-Lee-Marsh proved that the conformal compactification is in fact an intrinsic geometric notion, and managed to build conformal compactifications for asymptotically hyperbolic manifolds from their intrinsic inner geometry.
The complex counterpart of this is the notion of asymptotically complex hyperbolic manifolds, the analogue of conformal boundaries being the notion of CR boundaries (for Cauchy-Riemann). The geometry at infinity here is modelled on that of the complex hyperbolic space. In particular, such manifolds are asymptotically Einstein.
The CR compactification for such manifolds first arose as an extrinsic notion. In this talk, we prove that one can build CR compactifications for asymptotically complex hyperbolic manifolds out of the intrinsic geometric properties. The resulting boundary is a strictly pseudoconvex CR manifold.