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Properties of squeezing functions on $h$-extendible domains

Ninh Van Thu

TL;DR

The paper develops a scaling framework to analyze the boundary behavior of the squeezing function on bounded domains in C^{n+1}. By reducing local geometry to canonical models (Siegel half-spaces or their ℎ-extendible counterparts) and composing explicit biholomorphisms to the unit ball, it proves that the localized squeezing function tends to 1 near strongly pseudoconvex boundaries and along specific tangential approaches to finite-type boundaries in higher dimensions and in C^2. The main contributions are three theorems: (i) σ→1 at strongly pseudoconvex points (local version of known results); (ii) σ→1 along uniformly Λ-tangential sequences at strongly h-extendible points; and (iii) σ→1 along spherically 1/(2m)-tangential sequences in C^2, with counterexamples illustrating the necessity of hypotheses. Together, these results advance understanding of when the squeezing function detects strong pseudoconvexity and demonstrate the power of the Pinchuk scaling method in higher dimensions and finite-type settings.

Abstract

The purpose of this article is twofold. First, we prove that the squeezing function approaches 1 near strongly pseudoconvex boundary points of bounded domains in $\mathbb{C}^{n+1}$. Second, we show that the squeezing function approaches 1 along certain sequences converging to pseudoconvex boundary points of finite type, including uniformly $Λ$-tangential and spherically $\frac{1}{2m}$-tangential convergence patterns.

Properties of squeezing functions on $h$-extendible domains

TL;DR

The paper develops a scaling framework to analyze the boundary behavior of the squeezing function on bounded domains in C^{n+1}. By reducing local geometry to canonical models (Siegel half-spaces or their ℎ-extendible counterparts) and composing explicit biholomorphisms to the unit ball, it proves that the localized squeezing function tends to 1 near strongly pseudoconvex boundaries and along specific tangential approaches to finite-type boundaries in higher dimensions and in C^2. The main contributions are three theorems: (i) σ→1 at strongly pseudoconvex points (local version of known results); (ii) σ→1 along uniformly Λ-tangential sequences at strongly h-extendible points; and (iii) σ→1 along spherically 1/(2m)-tangential sequences in C^2, with counterexamples illustrating the necessity of hypotheses. Together, these results advance understanding of when the squeezing function detects strong pseudoconvexity and demonstrate the power of the Pinchuk scaling method in higher dimensions and finite-type settings.

Abstract

The purpose of this article is twofold. First, we prove that the squeezing function approaches 1 near strongly pseudoconvex boundary points of bounded domains in . Second, we show that the squeezing function approaches 1 along certain sequences converging to pseudoconvex boundary points of finite type, including uniformly -tangential and spherically -tangential convergence patterns.

Paper Structure

This paper contains 14 sections, 6 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal convergence
  4. Catlin's multitype
  5. The $h$-extendibility
  6. The boundary behavior of the squeezing function near a strongly pseudoconvex point
  7. Geometry of strongly pseudoconvex hypersurfaces
  8. Proof of Theorem \ref{['maintheorem1']}
  9. The boundary behavior of the squeezing function near a strongly $h$-extendible point
  10. $\Lambda$-tangential convergence
  11. Proof of Theorem \ref{['maintheorem2']}
  12. The boundary behavior of the squeezing function near a weakly pseudoconvex point in $\mathbb C^2$
  13. The spherically tangential convergence
  14. Proof of Theorem \ref{['maintheorem3']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n+1}$ and $\xi_0\in \partial \Omega$. If $\partial \Omega$ is $\mathcal{C}^2$ strongly pseudoconvex at $\xi_0$, then we have $\lim\limits_{z\to \xi_0}\sigma_{\Omega\cap U_0}(z)=1$, where $U_0$ is a sufficiently small neighborhood of $\xi_0$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 17 more