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On CR maps from the sphere into the tube over the future light cone II: Higher dimensions

Michael Reiter, Duong Ngoc Son

Abstract

We determine all CR maps from the sphere in $\mathbb{C}^3$ into the tube over the future light cone in $\mathbb{C}^4$. This result leads to a complete characterization of proper holomorphic maps from the three-dimensional unit ball into the classical domain of type IV of four dimension and confirms a conjecture of Reiter - Son in [26] from 2022. Additionally, we prove a boundary characterization of isometric holomorphic embeddings from a ball into a classical domain of type IV in arbitrary dimensions that is similar to the main result in Huang - Lu - Tang - Xiao [16]. The result is then used to treat a special case in the general characterization.

On CR maps from the sphere into the tube over the future light cone II: Higher dimensions

Abstract

We determine all CR maps from the sphere in into the tube over the future light cone in . This result leads to a complete characterization of proper holomorphic maps from the three-dimensional unit ball into the classical domain of type IV of four dimension and confirms a conjecture of Reiter - Son in [26] from 2022. Additionally, we prove a boundary characterization of isometric holomorphic embeddings from a ball into a classical domain of type IV in arbitrary dimensions that is similar to the main result in Huang - Lu - Tang - Xiao [16]. The result is then used to treat a special case in the general characterization.

Paper Structure

This paper contains 13 sections, 18 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The tube over the future light cone
  4. Stability groups
  5. Normalization, geometric rank, and isometric embeddings
  6. Normalization
  7. Geometric rank and the CR Ahlfors tensor
  8. Isometric embeddings
  9. Proof of Theorem \ref{['main']}
  10. Case 1:
  11. Case 2:
  12. Nowhere transversal maps
  13. Construction of maps, proof of Corollary \ref{['cor:balld4']}, and an example

Key Result

Theorem \oldthetheorem

Let $U$ be an open subset of $\mathbb{H}^5$ and $H\colon U \to \mathcal{X} \subset \mathbb{C}^4$ a $C^2$-smooth CR map. Then the following hold:

Theorems & Definitions (36)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Definition 3.1
  • Definition 3.2
  • Proposition \oldthetheorem: Lamel--Son lamel2019cr
  • Lemma \oldthetheorem
  • ...and 26 more