A complete normal form for everywhere Levi–degenerate hypersurfaces in C3

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

“…is in the right kernel of 𝐻 (i.e., 𝐻𝑣 = 0). A matrix representing one of the Levi kernel adjoint operators of 𝑀 (respectively, 𝑀 𝑗 ) is given by ( 8) with 𝑣 as in (17) (respectively, (15)), from which it is clearly seen that the matrix is a direct sum of matrices representing nonzero Levi kernel adjoint operators of 𝑀 1 and 𝑀 2 . Hence, 𝑀 = {ℜ(𝑧 0 ) = Φ} is uniformly 2-nondegenerate with a rank 1 Levi kernel.…”

“…The latter have been a subject of intensive research over recent decades, since [11,12] and the inception of the 𝑙-nondegeneracy notion in [2]. We refer to, for example, [4,5,7,8,13,14,21,23] (see [22] for the published version of [23]), and the introduction in [15] for a historic outline in the lowest dimension 𝑁 = 3 where 2-nondegeneracy occurs, and emphasize that in higher dimensions 𝑁 > 3 the study of 2-nondegenerate hypersurfaces exhibits considerable further difficulties (compared to the 𝑁 = 3 case). Principally, in higher dimensions the structures have nontrivial CR symbols, a basic local invariant.…”

“…A surprising dearth of nonhomogeneous constant symbol CR structure examples (of the type considered in [24,27]) was this work's initial motivation. Indeed, although there are general arguments ensuring existence of a rich variety of such structures, for example, such variety becomes apparent when contrasting the main results of [8] and [15], relatively few hypersurface realizations of such structures have been described in a closed form before.…”