On 3-nondegenerate CR manifolds in dimension 7 (II): the intransitive case

TL;DR

This paper studies 3-nondegenerate CR structures in 7 dimensions, establishing bounds on their symmetry algebra dimensions and demonstrating the existence of models with specific symmetry properties, advancing understanding of CR geometry.

Contribution

It provides new bounds on the symmetry algebra dimensions for 3-nondegenerate CR structures in dimension 7 and constructs models with maximal and submaximal symmetries.

Findings

Maximal symmetry algebra dimension is 8.

Next possible symmetry dimension is 6.

Existence of infinitely many non-equivalent submaximally symmetric models.

Abstract

We investigate 3-nondegenerate CR structures in the lowest possible dimension 7 and show that 8 is the maximal dimension for the Lie algebra of symmetries of such structures. The next possible symmetry dimension is 6, and for the automorphism groups the dimension 7 is also realizable. This part (II) is devoted to the case where the symmetry algebra acts intransitively. We use various methods to bound its dimension and demonstrate the existence of infinitely many non-equivalent submaximally symmetric models. Summarizing, we get a stronger form of Beloshapka's conjecture on the symmetry dimension of hypersurfaces in $\mathbb{C}^4$.