Alan Pinoy: CR compactification for asymptotically complex hyperbolic Hermitian manifolds

The complex hyperbolic space is the unique simply connected, complete Kähler manifold whose holomorphic sectional curvature is constant equal to -1. It is the complex analogue of the real hyperbolic space. It is biholomorphic to the unit ball of C^N, and can be compactified by a sphere at infinity. As a real hypersurface of C^N, this sphere at infinity carries a natural geometry, called a Cauchy-Riemann (CR) structure. One can recover abstractly this structure by analysing the expansion of the complex hyperbolic metric near infinity.
In this talk, we will consider a complete, non-compact Hermitian manifold, whose geometry at infinity is close to that of the complex hyperbolic space. Under natural geometric conditions, we construct a compactification having all the desired features: as in the model case, the boundary is a strictly pseudoconvex CR manifold, whose structure naturally appears in the expansion of the metric. This result is an extension of previous results I obtained during my PhD in the Kähler setting.