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Removable sets for pseudoconvexity for weakly smooth boundaries

Quang Dieu Nguyen, Pascal J. Thomas

TL;DR

The work investigates removability of pseudoconvexity for bounded domains in $\mathbb{C}^n$ with $\mathcal{C}^{1,1}$ boundaries, showing that a closed set $F\subset \partial\Omega$ of zero $(2n-1)$-dimensional measure, whose complement is $\mathcal{C}^2$-smooth and locally pseudoconvex, suffices to imply global pseudoconvexity. It develops a structure-hypotheses framework based on peak plurisubharmonic functions to produce a negative psh exhaustion and hyperconvexity, yielding GR-type removability results under minimal boundary regularity. The paper further derives a local Levi-condition criterion in dimension two, expressed as $-\Delta_{\tau(\varphi)}\varphi \ge 0$ on the relevant region, and proves local pseudoconvexity under weakened Sobolev and Hölder regularity assumptions, then lifts the argument to higher dimensions via slicing. It also provides sharp counterexamples showing the necessity of stronger hypotheses in low-regularity settings and discusses Zygmund regularity and Hausdorff-dimension effects, illustrating limits of removability in diverse regularity classes. Overall, the results sharpen the understanding of how boundary regularity and marginal exceptional sets control pseudoconvexityRemovability and offer new proofs of classical GR-type results under minimal assumptions.

Abstract

We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial Ω\setminus F$ is $\mathcal C^{2}$-smooth and locally pseudoconvex at every point, then $Ω$ is globally pseudoconvex. Unlike in the globally $\mathcal C^{2}$-smooth case, the condition ``$F$ of (relative) empty interior'' is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.

Removable sets for pseudoconvexity for weakly smooth boundaries

TL;DR

The work investigates removability of pseudoconvexity for bounded domains in with boundaries, showing that a closed set of zero -dimensional measure, whose complement is -smooth and locally pseudoconvex, suffices to imply global pseudoconvexity. It develops a structure-hypotheses framework based on peak plurisubharmonic functions to produce a negative psh exhaustion and hyperconvexity, yielding GR-type removability results under minimal boundary regularity. The paper further derives a local Levi-condition criterion in dimension two, expressed as on the relevant region, and proves local pseudoconvexity under weakened Sobolev and Hölder regularity assumptions, then lifts the argument to higher dimensions via slicing. It also provides sharp counterexamples showing the necessity of stronger hypotheses in low-regularity settings and discusses Zygmund regularity and Hausdorff-dimension effects, illustrating limits of removability in diverse regularity classes. Overall, the results sharpen the understanding of how boundary regularity and marginal exceptional sets control pseudoconvexityRemovability and offer new proofs of classical GR-type results under minimal assumptions.

Abstract

We show that for bounded domains in with smooth boundary, if there is a closed set of -Lebesgue measure such that is -smooth and locally pseudoconvex at every point, then is globally pseudoconvex. Unlike in the globally -smooth case, the condition `` of (relative) empty interior'' is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.
Paper Structure (9 sections, 11 theorems, 76 equations)

This paper contains 9 sections, 11 theorems, 76 equations.

Table of Contents

  1. Introduction and results
  2. Use of structure hypotheses
  3. The case of dimension $2$
  4. Local expression of the Levi condition.
  5. Local result.
  6. The higher dimensional case
  7. Examples
  8. Proof of Proposition \ref{['c11nonpc']}.
  9. Zygmund regularity and Hausdorff dimension.

Key Result

Theorem 1.1

Let $\Omega= \{z\in\mathbb C^n: \rho(z)<0\}$ be a bounded domain, where $\rho$ is a continuous real-valued function on $\mathbb C^n$ such that there is a neighborhood $\mathcal{V}$ of $\{ \rho(z)=0\}= \partial \Omega$ with $\rho \in \mathcal{C}^{1,1}(\mathcal{V})$ and $|\nabla \rho (z)|\ge c_0 >0$, Then $\Omega$ is pseudoconvex.

Theorems & Definitions (17)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 7 more