2019/12/03 by Zhangchi Chen, Wei-Guo Foo, Chen, Zhangchi +4
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1912.01655
openalex publication_date 2019/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a 2-nondegenerate constant Levi rank 1 rigid Cω hypersurface M5 ⊂ ℂ3 in coordinates (z, ζ, w = u + iv): u = F(z,ζ,z,ζ). The Gaussier-Merker model u=\fraczz+ (1)/(2)z2ζ+(1)/(2) z2 ζ1-ζζ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone \x12+x22-x32=0\. Another representation is the tube u=(x2)/(1-y). Inspired by Alexander Isaev, we study rigid biholomorphisms: (z,ζ,w) \longmapsto ( f(z,ζ), g(z,ζ), ρ w+h(z,ζ) ) =: (z',ζ',w'). The G-M model has 7-dimensional rigid automorphisms group. A Cartan-type reduction to an e-structure was done by Foo-Merker-Ta in 1904.02562. Three relative invariants appeared: V0, I0 (primary) and Q0 (derived). In Pocchiola's formalism, Section 8 provides a finalized expression for Q0. The goal is to establish the Poincaré-Moser complete normal form: u = \fraczz+(1)/(2) z2ζ +(1)/(2) z2ζ 1-ζζ + ∑a,b,c,d \atop a+c\geqslant 3 Ga,b,c,d zaζbzcζd, with 0 = Ga,b,0,0 = Ga,b,1,0 = Ga,b,2,0 and 0 = G3,0,0,1 = \rm Im G3,0,1,1. We apply the method of Chen-Merker 1908.07867 to catch (relative) invariants at every point, not only at the central point, as the coefficients G0,1,4,0, G0, 2, 3, 0, \rm Re G3,0,1,1. With this, a brige Poincaré \longleftrightarrow Cartan is constructed. In terms of F, the numerators of V0, I0, Q0 incorporate 11, 52, 824 differential monomials.