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Visibility domains that are not pseudoconvex

Annapurna Banik

TL;DR

The paper addresses whether every visibility domain with respect to the Kobayashi distance must be pseudoconvex. It demonstrates the existence of non-pseudoconvex, hence not Kobayashi complete, visibility domains by removing carefully chosen sets from a bounded strongly pseudoconvex domain: first a finite set, then a broader class of compact sets $A$ with $\mathcal{H}^{2n-2}(A)=0$ under a Lipschitz-path constraint. A corollary uses pseudo-arcs to produce explicit $A$ and shows the general construction applies widely. By leveraging almost-geodesics, Hartogs phenomena, and boundary metric estimates, the work decouples visibility from pseudoconvexity and completeness, expanding the zoo of visibility domains and clarifying the geometric limits of current intuition.

Abstract

The earliest examples of visibility domains, given by Bharali--Zimmer, are pseudoconvex. In fact, all known examples of visibility domains are pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.

Visibility domains that are not pseudoconvex

TL;DR

The paper addresses whether every visibility domain with respect to the Kobayashi distance must be pseudoconvex. It demonstrates the existence of non-pseudoconvex, hence not Kobayashi complete, visibility domains by removing carefully chosen sets from a bounded strongly pseudoconvex domain: first a finite set, then a broader class of compact sets with under a Lipschitz-path constraint. A corollary uses pseudo-arcs to produce explicit and shows the general construction applies widely. By leveraging almost-geodesics, Hartogs phenomena, and boundary metric estimates, the work decouples visibility from pseudoconvexity and completeness, expanding the zoo of visibility domains and clarifying the geometric limits of current intuition.

Abstract

The earliest examples of visibility domains, given by Bharali--Zimmer, are pseudoconvex. In fact, all known examples of visibility domains are pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.
Paper Structure (5 sections, 5 theorems, 13 equations)

This paper contains 5 sections, 5 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Essential lemmas and definitions
  3. The proof of Theorem \ref{['th:non-pscvx_Haus-Lip']}
  4. The proofs of Theorem \ref{['th:non-pscvx_basic']} and Corollary \ref{['cor:non-pscvx_simple_Haus-Lip']}
  5. Some questions

Key Result

Theorem 1.2

Let $D \subset \mathbb{C}^n$, $n\geq 2$, be a bounded domain with $\mathcal{C}^2$-smooth boundary and assume that $D$ is strongly Levi pseudoconvex. Let $A$ be a finite subset of $D$. Then, $\Omega:= D \setminus A$ is a visibility domain that is not pseudoconvex. In particular, $\Omega$ is not Kobay

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Corollary 1.5
  • Definition 2.1
  • Lemma 2.2: Bharali--Zimmer, bharali-zimmer:2017
  • Lemma 2.3
  • Definition 2.4: Bharali--Zimmer, bharali-zimmer:2023
  • proof : Proof of Theorem \ref{['th:non-pscvx_basic']}
  • ...and 1 more