On the Boundary Behaviour of Invariants and Curvatures of the Kobayashi--Fuks Metric in Strictly Pseudoconvex Domains

TL;DR

The paper analyzes the boundary behaviour of the Kobayashi-Fuks metric on bounded $C^2$-smooth strictly pseudoconvex domains using scaling to avoid localization of invariants. Through a Ramadanov-type stability result for the Bergman kernel and a Pinchuk-style holomorphic change of coordinates plus anisotropic scaling, the authors connect general domains to a model Siegel half-space and derive precise boundary limits for the metric and its invariants, including the canonical invariant $\beta$, holomorphic sectional curvature $R$, and Ricci curvature $\operatorname{Ric}$. The results yield explicit normal and tangential asymptotics near boundary points, establish local stability, and show isometry rigidity: isometries of the Kobayashi-Fuks metric are holomorphic or conjugate-holomorphic. These findings extend prior work by refining scaling methods and providing a unified growth-profile for invariants at the boundary, with explicit constants in the model and ball cases.

Abstract

The purpose of this article is to investigate the boundary behaviour of the Kobayashi--Fuks metric and several associated invariants on strictly pseudoconvex domains in the paradigm of scaling. This approach allows us to examine more invariants, such as the canonical invariant, holomorphic sectional curvature, and Ricci curvature of this metric, in a manner that extends and refines some existing analysis.

Theorems & Definitions (18)

• Example 1
• Example 2
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• proof : Proof of Lemma \ref{['ram']}