Group-Invariant CR maps of the sphere $\mathbb{S}^5$

TL;DR

This work extends the theory of group-invariant CR maps to the sphere $\mathbb{S}^5\subset\mathbb{C}^3$ by classifying admissible subgroups $\Gamma\subset U(3)$ and constructing sharp gap-termination bounds for the minimal embedding dimension. It leverages the polynomial-positivity framework: a canonical $\Gamma$-invariant polynomial $f_{\Gamma}$ of rank $N(\Gamma)$, together with iterates under maps $F_j,G_j,H$, yields families of $\Gamma$-invariant polynomials $f_{m_0,m}$ with controllable ranks. The main result provides explicit thresholds $n(\Gamma)$ for four subgroup types, demonstrating that for $N\ge n(\Gamma)$ there exist smooth nonconstant $\Gamma$-invariant CR maps $\mathbb{S}^5\to\mathbb{S}^{2N-1}$ realizing $N$ as the minimal embedding dimension; in particular, the bounds are concrete and comparable to the sharp bounds known in lower dimensions. Collectively, these findings advance understanding of gap phenomena in higher-dimensional CR geometry and suggest similar rank-augmentation strategies may apply to broader settings.

Abstract

We construct group-invariant CR maps from the unit sphere in $\mathbb{C}^3$ and provide sharp bounds for the gap termination in this setting.

Theorems & Definitions (11)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• Corollary 2.5
• proof : Proof of Lemma \ref{['lemran1']}
• Lemma 2.6
• proof
• Lemma 2.7
• proof