# Lower bounds on the Bergman metric near points of infinite type

## TL;DR

This paper establishes a lower bound for the Bergman metric near boundary points of infinite type in pseudoconvex domains, refining previous results by reducing boundary smoothness requirements.

## Contribution

It provides a new lower bound for the Bergman metric in pseudoconvex domains with an $f$-property, improving upon prior work by Khanh and Zampieri with less smoothness assumptions.

## Key findings

- Derived a lower bound for the Bergman metric involving a specific function $ ilde g$
- Extended the analysis to points of infinite type on the boundary
- Reduced the boundary smoothness assumptions needed for the bounds

## Abstract

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$ satisfying an $f$-property for some function $f$. We show that the Bergman metric associated to $\Omega$ has the lower bound $\tilde g(\delta_\Omega(z)^{-1})$ where $\delta_\Omega(z)$ is the distance from $z$ to the boundary $\partial\Omega$ and $\tilde g$ is a specific function defined by $f$. This refines Khanh-Zampieri's work in \cite{KZ12} with reducing the smoothness assumption of the boundary.