Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The union problem for domains with partial pseudoconvex boundaries

Jinjin Hu, Xujun Zhang

TL;DR

The paper studies the union problem for bounded domains $D\subset \mathbb{C}^n$ with smooth boundaries by proving that a domain expressible as an increasing union $D=\bigcup_j D_j$ of hyper-$q$-convex subdomains is itself hyper-$q$-convex. A new characterization of hyper-$q$-convexity is established via solvability and $L^2$-estimates for weighted $\bar{\partial}$-problems with weights $\varphi=a\|z-z_0\|^2-b$, together with a boundary-type Bochner inequality. The approach combines Hörmander-type $L^2$ existence results, localization and a delicate boundary analysis to rule out non-hyper-$q$-convex boundary points. Additionally, a convex-analytic (real) analogue is developed using a weighted $d$-complex, yielding a parallel union result for $q$-convexity and highlighting the parallel between complex-analytic and real differential-geometric convexity.

Abstract

We show that a smooth bounded domain in $\mathbb{C}^n$ admitting partial pseudoconvex exhaustion remains partial pseudoconvex. The main ingredient of the proof is based on a new characterization of hyper-$q$-convex domains. Furthermore, we get several convex analogies.

The union problem for domains with partial pseudoconvex boundaries

TL;DR

The paper studies the union problem for bounded domains with smooth boundaries by proving that a domain expressible as an increasing union of hyper--convex subdomains is itself hyper--convex. A new characterization of hyper--convexity is established via solvability and -estimates for weighted -problems with weights , together with a boundary-type Bochner inequality. The approach combines Hörmander-type existence results, localization and a delicate boundary analysis to rule out non-hyper--convex boundary points. Additionally, a convex-analytic (real) analogue is developed using a weighted -complex, yielding a parallel union result for -convexity and highlighting the parallel between complex-analytic and real differential-geometric convexity.

Abstract

We show that a smooth bounded domain in admitting partial pseudoconvex exhaustion remains partial pseudoconvex. The main ingredient of the proof is based on a new characterization of hyper--convex domains. Furthermore, we get several convex analogies.

Paper Structure

This paper contains 9 sections, 14 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations for the $\bar{\partial}$ complex
  4. Notations for variants of partial convexity
  5. Characterize the domain with smooth partial pseudoconvex boundary
  6. The Union Problem for the hyper-$q$-convex domain
  7. Convex Analogy
  8. The weighted $d$-complex
  9. Proof of Theorem \ref{['thm:main-3']}

Key Result

Theorem 1.1

Let $D$ be a bounded domain in $\mathbb{C}^n$ with a smooth boundary. Let $\{D_j\}$ be a sequence of open subsets of $D$ with $D_j\subset D_{j+1}$ and $\bigcup_jD_j=D$. If each $D_j$ is hyper-$q$-convex, then $D$ is hyper-$q$-convex.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Definition 2.1: Grauert-Riemenschneider,SYT-1982
  • Theorem 2.2: Ho-1991
  • Lemma 2.3: Ho-1991Kohn63
  • Lemma 3.1: Deng-Zhang
  • Theorem 3.2
  • ...and 9 more