Einstein metrics on strictly pseudoconvex domains from the viewpoint of bulk-boundary correspondence

TL;DR

This survey explores the relationship between asymptotically complex hyperbolic Einstein metrics and CR structures on boundaries, highlighting existence results and open problems in the complex bulk-boundary correspondence.

Contribution

It provides a comprehensive overview of existence theorems and open questions in the complex analogue of bulk-boundary correspondence for Einstein metrics.

Findings

Summarizes key existence theorems for complex hyperbolic Einstein metrics.

Identifies open problems in the field.

Clarifies the analogy between Poincaré-Einstein and complex hyperbolic metrics.

Abstract

This is a survey on the correspondence between asymptotically complex hyperbolic Einstein metrics and CR structures on the boundary at infinity, which is the complex version of that between Poincar\'e-Einstein metrics and conformal structures. We mainly discuss existence theorems, and propose several open problems.