Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic behavior of the Bergman kernel and associated invariants in weakly pseudoconvex domains

Ninh Van Thu

TL;DR

This work extends the Fefferman-type boundary analysis of Bergman-type invariants to weakly pseudoconvex domains by developing scaling methods anchored in Catlin's multitype and $h$-extendibility. It establishes explicit asymptotics for the Bergman kernel on the diagonal, the Bergman metric, and associated curvatures along sequences converging to strongly $h$-extendible or finite-type boundary points, using uniform $oldsymbol{ extLambda}$-tangential and spherical $ rac{1}{2m}$-tangential convergences. The authors derive model-domain limits via anisotropic scalings that reduce to standard domains (Siegel half-spaces and the unit ball), and they prove localization of minimum integrals to transfer local boundary geometry into global invariant estimates. Key results include precise formulas for $K_ ext{Ω}(oldeta_j,oldeta_j)$, $d^2_ ext{Ω}(oldeta_j;oldsymbol{ extxi})$, and curvature limits as $j o\infty$, showing convergence to unit-ball values under the right tangential regimes. The findings illuminate how boundary geometry (strongly $h$-extendible vs. weakly pseudoconvex) governs the asymptotic behavior of Bergman invariants, with implications for several complex variables and several complex geometric analysis.

Abstract

In this paper, we present an explicit description for the boundary behavior of the Bergman kernel function, the Bergman metric, and the associated curvatures along certain sequences converging to an $h$-extendible boundary point.

Asymptotic behavior of the Bergman kernel and associated invariants in weakly pseudoconvex domains

TL;DR

This work extends the Fefferman-type boundary analysis of Bergman-type invariants to weakly pseudoconvex domains by developing scaling methods anchored in Catlin's multitype and -extendibility. It establishes explicit asymptotics for the Bergman kernel on the diagonal, the Bergman metric, and associated curvatures along sequences converging to strongly -extendible or finite-type boundary points, using uniform -tangential and spherical -tangential convergences. The authors derive model-domain limits via anisotropic scalings that reduce to standard domains (Siegel half-spaces and the unit ball), and they prove localization of minimum integrals to transfer local boundary geometry into global invariant estimates. Key results include precise formulas for , , and curvature limits as , showing convergence to unit-ball values under the right tangential regimes. The findings illuminate how boundary geometry (strongly -extendible vs. weakly pseudoconvex) governs the asymptotic behavior of Bergman invariants, with implications for several complex variables and several complex geometric analysis.

Abstract

In this paper, we present an explicit description for the boundary behavior of the Bergman kernel function, the Bergman metric, and the associated curvatures along certain sequences converging to an -extendible boundary point.

Paper Structure

This paper contains 15 sections, 10 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal convergence
  4. Catlin's multitype
  5. The $h$-extendibility
  6. The Bergman kernel, the Bergman metric, and its curvatures
  7. The minimum integrals
  8. Localization of minimum integrals
  9. The boundary behavior of the Bergman kernel function, the Bergman metric, and the associated curvatures
  10. The boundary behavior of the Bergman kernel, the Bergman metric, and curvatures near a strongly $h$-extendible point
  11. $\Lambda$-tangential convergence
  12. Estimates of Bergman kernel function and associated invariants near a strongly $h$-extendible point
  13. The boundary behavior of the Bergman kernel, the Bergman metric, and curvatures near a weakly pseudoconvex boundary point in $\mathbb C^2$
  14. The spherically tangential convergence
  15. Estimates of Bergman kernel function and associated invariants near a weakly pseudoconvex boundary point in $\mathbb C^2$

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n+1}$ with $C^\infty$-smooth boundary and $\xi_0\in\partial \Omega$ be strongly $h$-extendible with Catlin's finite multitype $(2m_1,\ldots,2m_n, 1)$ (cf. Definition strongly-h-extendible). Denote by $\Lambda=(1/2m_1, \ldots, 1/2m_n)$. If $\{\eta_j=(\ where $K_{\Omega}(p, p), d^2_\Omega(p;\xi),\mathrm{Sec}_{\Omega}(p,\xi),\mathrm{Ric}_{\Omega}(p,\x

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 23 more