Analysis Seminar

Date & Time: 10:00 am, Thursday, December 5, 2024

Speaker: Ravi Jaiswal, TIFR-CAM

Meeting link: https://meet.google.com/twn-hsiu-mze

*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 behaviour 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.

Finally, I will discuss some future research plans in the directions
mentioned above.