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Taut visibility domains are not necessarily Kobayashi complete

Rumpa Masanta

TL;DR

The paper resolves Banik's question by constructing, for each $n\ge 2$, a bounded domain $\Omega\subset \mathbb{C}^n$ with $0\in\partial\Omega$, smooth boundary away from $0$, that is taut and a visibility domain with respect to the Kobayashi distance, yet $(\Omega,K_\Omega)$ is not Cauchy-complete. The construction blends a plurisubharmonic exhaustion on a product domain, a carefully chosen set $X$, a sequence $(x_\nu)$, and a PSH function $u$ to define $\Omega$, ensuring tautness and visibility through local Goldilocks points, while producing a $K_\Omega$-Cauchy sequence that does not converge in $\Omega$. This demonstrates that taut visibility does not imply Kobayashi completeness, sharpening the understanding of Kobayashi geometry in several complex variables and providing explicit counterexamples across dimensions.

Abstract

We answer a question asked recently by Banik in the negative by showing that for each $n\geq 2$, there exists a taut visibility domain in $\mathbb{C}^n$ that is not Kobayashi complete. The domains that we produce are bounded and have boundaries that are very regular away from a single point.

Taut visibility domains are not necessarily Kobayashi complete

TL;DR

The paper resolves Banik's question by constructing, for each , a bounded domain with , smooth boundary away from , that is taut and a visibility domain with respect to the Kobayashi distance, yet is not Cauchy-complete. The construction blends a plurisubharmonic exhaustion on a product domain, a carefully chosen set , a sequence , and a PSH function to define , ensuring tautness and visibility through local Goldilocks points, while producing a -Cauchy sequence that does not converge in . This demonstrates that taut visibility does not imply Kobayashi completeness, sharpening the understanding of Kobayashi geometry in several complex variables and providing explicit counterexamples across dimensions.

Abstract

We answer a question asked recently by Banik in the negative by showing that for each , there exists a taut visibility domain in that is not Kobayashi complete. The domains that we produce are bounded and have boundaries that are very regular away from a single point.
Paper Structure (3 sections, 3 theorems, 3 equations)

This paper contains 3 sections, 3 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem \ref{['T:main-result']}

Key Result

Theorem 1.1

For each $n\geq 2$, there exists a bounded domain $\Omega\varsubsetneq\mathbb{C}^n$ with the properties such that $\Omega$ is taut and is a visibility domain with respect to the Kobayashi distance, but $(\Omega,K_\Omega)$ is not Cauchy-complete.

Theorems & Definitions (7)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 2.1
  • Proposition 2.2
  • proof : The proof of Theorem \ref{['T:main-result']}
  • Remark 3.1