Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On 3-nondegenerate CR manifolds in dimension 7 (I): the transitive case

Boris Kruglikov, Andrea Santi

Abstract

We investigate 3-nondegenerate CR structures in the lowest possible dimension 7, and one of our goals is to prove Beloshapka's conjecture on the symmetry dimension bound for hypersurfaces in $\mathbb{C}^4$. We claim that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model. This part (I) is devoted to the homogeneous case: we prove that the model is locally the only homogeneous 3-nondegenerate CR structure in dimension 7.

On 3-nondegenerate CR manifolds in dimension 7 (I): the transitive case

Abstract

We investigate 3-nondegenerate CR structures in the lowest possible dimension 7, and one of our goals is to prove Beloshapka's conjecture on the symmetry dimension bound for hypersurfaces in . We claim that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model. This part (I) is devoted to the homogeneous case: we prove that the model is locally the only homogeneous 3-nondegenerate CR structure in dimension 7.
Paper Structure (22 sections, 22 theorems, 134 equations)

This paper contains 22 sections, 22 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Freeman filtration and universal CR algebras
  3. The filtration of Freeman
  4. Locally homogeneous CR manifolds and CR algebras
  5. A filtration on the Lie algebra of infinitesimal CR automorphisms
  6. Universal CR algebra
  7. Infinitesimal CR automorphisms of Levi degenerate CR hypersurfaces
  8. From filtrations to gradings
  9. Trivial filtered deformations
  10. $k$-nondegenerate homogeneous models
  11. Homogeneous $7$-dimensional $3$-nondegenerate CR manifolds
  12. The maximal symmetric homogeneous space: abstract model
  13. The maximal symmetric homogeneous space in disguise
  14. Proof of the main results: local theory
  15. Global models
  16. ...and 7 more sections

Key Result

Theorem 2

Let $(M ,\mathcal{D},\mathcal{J})$ be a 3-nondegenerate 7-dimensional real-analytic connected CR-hypersurface. Assume the symmetry algebra $\mathfrak{g}$ acts with an open orbit.This assumption is not essential and can be removed in light of our follow-up paper KS2 devoted to the intransitive case.

Theorems & Definitions (52)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Lemma 7
  • proof
  • Proposition 8
  • proof
  • ...and 42 more