2023/10/28 by Jan Gregorovič, Gregorovič, Jan, David Sykes +1 · 1 citation
Mathematics · #32V05 #32V40 #53C30 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2310.18588
openalex publication_date 2023/10/28 · openalex created_date 2023/11/01 · openalex updated_date 2026/08/01
We study CR hypersurfaces in ℂ4 that are Levi degenerate with constant rank Levi form, and moreover finitely nondegenerate. Each of these can be described as a deformation of a model CR hypersurface by adding terms of higher natural weighted order to the model's defining equation. We obtain a complete normal form for models of real analytic uniformly 2-nondegenerate CR hypersurfaces in ℂ4, and present a detailed study of their local invariants. The normal form illustrates that 2-nondegenerate models in ℂ4 comprise a moduli space parameterized by two univariate holomorphic functions, which is in sharp contrast to the well known Levi-nondegenerate setting and the more recently discovered behavior of 2-nondegenerate structures in ℂ3. In further contrast to these previously studied settings, we demonstrate that not all 2-nondegenerate structures in ℂ4 arise as perturbations of homogeneous models. We derive defining equations for the homogeneous 2-nondegenerate models, a set of 9 structures, and find explicit formulas for their infinitesimal symmetries.