Ph.D Synopsis by Mr.Ravi Shankar Jaiswal

Abstract

We will prove optimal lower and upper bounds of the Bergman and Szegő kernels near the boundary of bounded smooth generalized decoupled pseudoconvex domains in ℂⁿ⁺¹. Generalized decoupled domains may have complex tangential directions that are not necessarily decoupled individually, and their boundary points may possess both finite and infinite type directions.

We will then proceed to study exponentially flat infinite type domains. On this class of domains, we will prove nontangential asymptotic limits of the following at exponentially flat infinite type boundary points of smooth bounded pseudoconvex domains in  ℂⁿ⁺¹: Bergman kernel and metric, Kobayashi and Kobayashi–Fuks metrics, holomorphic sectional, Ricci and scalar curvatures of the Bergman metric, and Bergman canonical invariant.