Some remarks on the Kobayashi--Fuks metric on strongly pseudoconvex domains

TL;DR

This paper investigates the properties of the Kobayashi--Fuks metric on strongly pseudoconvex domains, focusing on its localization near peak points and its similarities to the Bergman metric.

Contribution

It provides new insights into the localization behavior of the Kobayashi--Fuks metric and compares its properties with those of the Bergman metric on strongly pseudoconvex domains.

Findings

The Kobayashi--Fuks metric localizes near holomorphic peak points.

It shares several properties with the Bergman metric on strongly pseudoconvex domains.

The metric exhibits bounded curvature behavior similar to the Bergman metric.

Abstract

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a K\"ahler metric on $D$ known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.