Boundary behaviour of the Fefferman--Szegö metric in strictly pseudoconvex domains

TL;DR

This work addresses boundary behaviour of the Fefferman–Szegö metric on $C^\infty$-smoothly bounded strictly pseudoconvex domains. It employs Pinchuk scaling to prove Ramadanov-type stability theorems for the Szegö kernel and shows convergence to the unit-ball kernel under scaling, with the $SK_\Omega$ invariance linking Szegö and Bergman kernels. The authors derive explicit boundary limits for the fundamental invariants $g_\Omega$, $\beta_\Omega$, $ds_\Omega$, and curvature quantities $R_\Omega$ and $\operatorname{Ric}_\Omega$, matching the ball model and revealing precise normal/tangential asymptotics. Overall, the results illuminate the parallel between Fefferman–Szegö and Bergman metrics in the boundary regime and yield rigidity statements for isometries, enhancing understanding of complex-geometric analysis on strictly pseudoconvex domains.

Abstract

We study the boundary behaviour of the Fefferman--Szegö metric and several associated invariants in a $C^\infty$-smoothly bounded strictly pseudoconvex domain.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Theorem \ref{['ram-sgo']}
• proof : Proof of Theorem \ref{['ram-sgo-loc']}
• Lemma 3.1
• proof