# Deformation of Einstein metrics and almost complex structures on strictly pseudoconvex domains

# Deformation of Einstein metrics and almost complex structures on strictly pseudoconvex domains

I will discuss two problems on asymptotically complex hyperbolic spaces (ACH spaces): given a domain whose boundary is equipped with a contact distribution and a “compatible” almost CR structure, (I) construct an Einstein ACH metric whose conformal infinity is the given almost CR structure; (II) extend the almost CR structure to an almost complex structure of the domain in a preferable way. Our models are bounded strictly pseudoconvex domains in $\mathbb{C}^n$ equipped with Cheng-Yau’s complete Kähler-Einstein metric. I will present perturbative existence results for (I) and (II) that deform the Cheng-Yau situation. Problem (I) is a generalization of the ACH counterpart of the Graham-Lee theorem for AH metrics, while (II) is completely new.