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Envelope of truncated tubes and special domains in higher complex dimensions

Suprokash Hazra

TL;DR

The article develops a geometric framework, via special domains built from a convex initial base and a good barrier, to study envelopes of holomorphy for truncated tube-type domains in higher complex dimensions. It proves that every pseudoconvex truncated tube is a special domain and derives a sharp envelope theorem for these domains in $\mathbb{C}^n$, extending the Jarnicki–Pflug results, with explicit hull descriptions and a chain of analytic continuation arguments. A significant contribution is a higher-dimensional schlichtness result, complemented by a 2D refinement for domains with convex holes and two higher-dimensional generalizations using Stout’s theory. The work combines convexity-of-hulls, CR-geometry, and extension theorems to connect the base-domain geometry with the global envelope, and it poses several open questions about schlichtness criteria under broader geometric hypotheses.

Abstract

In this article, we introduce special domains and discuss the geometry of these domains, which includes showing that every pseudoconvex truncated tube domain is a special domain. Next, we prove a theorem for the envelope of special domains in $\C^n ~(n\geq 2)$. Our theorem on special domains is a generalization of a recent result by Jarnicki-Pflug on the envelope of holomorphy of truncated tube domains in $\C^n$. We also establish a result on schlichtness in complex dimension 2, and conclude this article with two higher-dimensional generalizations of the same result by Jarnicki-Pflug mentioned above.

Envelope of truncated tubes and special domains in higher complex dimensions

TL;DR

The article develops a geometric framework, via special domains built from a convex initial base and a good barrier, to study envelopes of holomorphy for truncated tube-type domains in higher complex dimensions. It proves that every pseudoconvex truncated tube is a special domain and derives a sharp envelope theorem for these domains in , extending the Jarnicki–Pflug results, with explicit hull descriptions and a chain of analytic continuation arguments. A significant contribution is a higher-dimensional schlichtness result, complemented by a 2D refinement for domains with convex holes and two higher-dimensional generalizations using Stout’s theory. The work combines convexity-of-hulls, CR-geometry, and extension theorems to connect the base-domain geometry with the global envelope, and it poses several open questions about schlichtness criteria under broader geometric hypotheses.

Abstract

In this article, we introduce special domains and discuss the geometry of these domains, which includes showing that every pseudoconvex truncated tube domain is a special domain. Next, we prove a theorem for the envelope of special domains in . Our theorem on special domains is a generalization of a recent result by Jarnicki-Pflug on the envelope of holomorphy of truncated tube domains in . We also establish a result on schlichtness in complex dimension 2, and conclude this article with two higher-dimensional generalizations of the same result by Jarnicki-Pflug mentioned above.

Paper Structure

This paper contains 14 sections, 26 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and some classical results
  3. Different hulls and some classical results
  4. Revisiting two classical results on envelope of holomorphy
  5. Geometry of compact fence and special domain
  6. Envelope of special domains in higher complex dimension
  7. Proof of the claim.
  8. Proof of the claim.
  9. Proof of the claim.
  10. Proof of the claim.
  11. Proof of the claim.
  12. Schlichtness of the envelope via Stout's theorem on analytic continuation
  13. Proof of the claim.
  14. Open questions

Key Result

Theorem 1.1

For $0\leq r_1 < r_2$ and $r_3>0$, the envelope of holomorphy of the truncated tube domain is given by In particular, the envelope is schlicht.

Figures (2)

  • Figure 2: Geometry of the domain $\textsf{Re}(S_{\alpha})$ in $\mathbb{R}^2$
  • Figure 3: Geometry of the domain $\mathcal{D}\subset \mathbb{C}^n$ in Theorem \ref{['2nd_theo']}.

Theorems & Definitions (63)

  • Theorem 1.1: Jarnicki-Pflug, JP2
  • Theorem 1.2: Hazra-Porten, haz-por
  • Definition 2.1: page 120, kra
  • Definition 2.2: CR manifold and CR function, bogg
  • Definition 2.3: Pseudoconvexity and Levi-flatness, kra
  • Theorem 2.4: bogg, chapter 10, Theorem 1
  • Theorem 2.5: Theorem 16, Chap. III, gun-ros
  • Lemma 2.6: Rosay-Stout, ros-st
  • Definition 2.7: stout, jor2
  • Theorem 2.8: Jöricke, jor2
  • ...and 53 more