Boundary Behaviour of Biholomorphic Invariants on Infinite Type Domains
Ravi Shankar Jaiswal (PhD Student, TIFR CAM)
| Speaker |
Ravi Shankar Jaiswal (PhD Student, TIFR CAM)
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|---|---|
| When |
Nov 28, 2024
from 04:00 PM to 05:00 PM |
| Where | LH-006, Ground Floor |
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PhD Thesis Defense
Title: Boundary Behaviour of Biholomorphic Invariants on Infinite Type Domains
Abstract: On domains in $\mathbb{C}^n$, $n > 1$, there is a deep interplay between the boundary geometry of the domain and function theory on the domain. The interplay is often captured in the boundary behaviour of various canonical objects associated to the domain, many of which are also biholomorphic invariants. Examining the boundary behavior of these objects provides insights into the behaviour of holomorphic mappings, and the classification of domains up to biholomorphic equivalence.
Motivated by the above facts, we will prove optimal lower and upper bounds of the Bergman and Szeg\H{o} kernels near the boundary of bounded smooth generalized decoupled pseudoconvex domains in $\mathbb{C}^{n}$. 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 $\mathbb{C}^{n}$: Bergman kernel and metric, Kobayashi and Kobayashi--Fuks metrics, holomorphic sectional, Ricci and scalar curvatures of the Bergman metric, and Bergman canonical invariant.