CR-manifolds of dimension 5: A Lie algebra approach

Abstract

Abstract. We study real-analytic Levi degenerate hypersurfaces M in complex manifolds of dimension 3, for which the CR-automorphism group Aut(M ) is a real Lie group acting transitively on M . We provide large classes of examples for such M , compute the corresponding groups Aut(M ) and determine the maximal subsets of M that cannot be separated by global continuous CR-functions. It turns out that all our examples, although partly arising in different contexts, are locally CR-equivalent to the tube T = C × i 3 ⊂ ¼ 3 over the future light cone3 , x 3 > 0} in 3-dimensional space-time.

Classification of Levi degenerate homogeneous CR-manifolds in dimension 5

“…Therefore, h: M !M is a CR-isomorphism. (ii) The proof is essentially the same as of Proposition 6.3 in [16].…”

“…In case (ii), M is a proper domain in M: we realize M=F +iR 3 in C 3 with coordinates (z 0 , z 1 , z 2 ) as F ={x∈R 3 :x 0 x 2 =x 2 1 and x 0 +x 2 >0}. Then hol(M) is the linear span of the vector fields (3.5) and (3.7) in [16]. We may assume without loss of generality that ϕ=ξ −1,1 =2z 1 ∂/∂z 0 +z 2 ∂/∂z 1 .…”

“…compare, e.g., [13], [16], [23] and [31]. This 5-dimensional CR-manifold has several remarkable properties and serves as motivation for various considerations in this paper.…”

“…So far, compare e.g. [14], [23], [16] and [5], all known examples in dimension 5 finally turned out to be locally CR-isomorphic to M. Even levi degenerate homogeneous cr-manifolds 3 the announced 'new example' in [17] revealed itself as locally equivalent to M as shown in [16]. Therefore the desire arose to find examples that are not locally CR-isomorphic to M.…”

Слоения И Глобализации Компактных Однородных CR-многообразий

“…Другие недавние результаты по данной тематике содержатся в работах Г. Фелса и В. Каупа (см., например, [8]). О классификации гиперповерхно-стей см., например, [2]- [4], [9]- [12].…”