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 C 4 . We claim that 8 is the maximal symmetry dimension of 3nondegenerate 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.

New examples of 2‐nondegenerate real hypersurfaces in CN$\mathbb {C}^N$ with arbitrary nilpotent symbols

“…We expect that many more such examples exist, although deriving their closed form descriptions will be a challenge as they arise from solutions to rather complicated overdetermined system of partial differential equations (PDEs). For 𝑘 > 2, there are examples of homogeneous structures (e.g., [6,8,9,16,18,20]) and the recent study [17] of nonhomogeneous 3-nondegenerate structures in ℂ 4 . Given the early state of this 𝑘 > 2 frontier, the analogous open problems of finding closed form defining equations are interesting for higher orders of 𝑘-nondegeneracy.…”