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

Abstract

We introduce a class of uniformly 2‐nondegenerate CR hypersurfaces in , for , having a rank 1 Levi kernel. The class is first of all remarkable by the fact that for every it forms an explicit infinite‐dimensional family of everywhere 2‐nondegenerate hypersurfaces. To the best of our knowledge, this is the first such construction. Besides, the class contains infinite‐dimensional families of nonequivalent structures having a given constant nilpotent CR symbol for every such symbol. Using methods that are able to handle all cases with simultaneously, we solve the equivalence problem for the considered structures whose symbol is represented by a single Jordan block, classify their algebras of infinitesimal symmetries, and classify the locally homogeneous structures among them. We show that the remaining considered structures, which have symbols represented by a direct sum of Jordan blocks, can be constructed from the single block structures through simple linking and extension processes.