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Truncated tube domains with multi-sheeted envelope

Suprokash Hazra, Egmont Porten

TL;DR

This paper investigates whether envelopes of holomorphy for truncated tube domains are schlicht and provides sharp counterexamples demonstrating multi-sheeted envelopes, including domains homeomorphic to the open $4$-ball with arbitrarily many sheets. The authors construct domains $X_k\Subset\mathbb{R}^2$ so that $D_k=X_k+iB_{r_k}$ have envelopes with at least $k$ sheets, and an $X_\infty$ yielding infinitely many sheets over a circle, using Levi extension and a spiral-geometry approach. In dimension two, they also establish a concrete schlichtness criterion: if $X$ is convex with finitely many strictly convex holes and $Y$ is convex, then $\textsf{E}(D)$ is schlicht, with explicit polynomial-hull descriptions in model cases. Together, the results delineate the boundary between schlicht and multi-sheeted envelopes and connect envelope structure to polynomial hulls and universal covering phenomena.

Abstract

The present article is concerned with a group of problems raised by J. Noguchi and M. Jarnicki/P. Plug, namely whether the envelopes of holomorphy of truncated tube domains are always schlicht, i.e. subdomains of $\mathbb{C^n}$, and how to characterize schlichtness if this is not the case. By way of a counter-example homeomorphic to the 4-ball, we answer the first question in the negative. Moreover, it is possible that the envelopes have arbitrarily many sheets. The article is concluded by sufficient conditions for schlichtness in complex dimension two.

Truncated tube domains with multi-sheeted envelope

TL;DR

This paper investigates whether envelopes of holomorphy for truncated tube domains are schlicht and provides sharp counterexamples demonstrating multi-sheeted envelopes, including domains homeomorphic to the open -ball with arbitrarily many sheets. The authors construct domains so that have envelopes with at least sheets, and an yielding infinitely many sheets over a circle, using Levi extension and a spiral-geometry approach. In dimension two, they also establish a concrete schlichtness criterion: if is convex with finitely many strictly convex holes and is convex, then is schlicht, with explicit polynomial-hull descriptions in model cases. Together, the results delineate the boundary between schlicht and multi-sheeted envelopes and connect envelope structure to polynomial hulls and universal covering phenomena.

Abstract

The present article is concerned with a group of problems raised by J. Noguchi and M. Jarnicki/P. Plug, namely whether the envelopes of holomorphy of truncated tube domains are always schlicht, i.e. subdomains of , and how to characterize schlichtness if this is not the case. By way of a counter-example homeomorphic to the 4-ball, we answer the first question in the negative. Moreover, it is possible that the envelopes have arbitrarily many sheets. The article is concluded by sufficient conditions for schlichtness in complex dimension two.
Paper Structure (4 sections, 7 theorems, 29 equations, 3 figures)

This paper contains 4 sections, 7 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. Universally covered circles
  4. Schlichtness in dimension $2$

Key Result

Theorem 1.1

For every integer $k\geq 2$ there is a domain $X_k\Subset\mathbb{R}^2$ with $\mathcal{C}^\infty$-smooth boundary and a radius $r_k>0$ such that the envelope of holomorphy of $D_k=X_k+iB_{r_k}$ has at least $k$ sheets over some point in $\mathbb{C}^2$. Furthermore, there are a domain $X_\infty\Subset

Figures (3)

  • Figure 2.1: Intersection of $D_{\infty}$ with $\mathbb{R}^2$
  • Figure 3.2: Intersection of $D$ with $\mathbb{R}^2$
  • Figure 4.3: The domain $X'$ in $\mathbb{R}^2$

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • Remark 4.2
  • ...and 3 more