Biholomorphic equivalence to totally nondegenerate model CR manifolds

Abstract

ApplyingÉlie Cartan's classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka's model of CR dimension one and codimension k > 1, whence of real dimension 2 + k, is reducible to some absolute parallelism, namely to an {e}-structure on a certain prolonged manifold of real dimension either 3 + k or 4 + k. The proof relies on the weight analysis of the structure equations associated with the mentioned problem of equivalence. Thanks to the achieved results, we prove Beloshapka's maximum conjecture about the rigidity of his CR models of certain lengths equal or greater than three: " CR automorphism Lie groups of these models do not contain any nonlinear map, preserving the origin ". Here, we mainly deal with CR models of the fixed CR dimension one though the results seem generalizable by means of certain analogous proofs.

On the maximum conjecture

“…Next in Section 4, we prove the conjecture in CR dimension one. As we already mentioned, the maximum conjecture in this CR dimension is actually proved before in [18] by means of Cartan's geometric approach for solving equivalence problems. Nevertheless, here we provide a much shorter proof using the parallel algebraic approach of Tanaka.…”

“…(cf. [18,Definition 1.1]). An arbitrary (local) real analytic generic CR manifold M is totally nondegenerate (or maximally minimal) of length ρ if the distribution D 1 = T c M is regular with the minimum possible degree of nonholonomy ρ.…”

“…Corresponding to (4), one may write: If this conjecture holds true, then Beloshapka's models of length ρ ≥ 3 never have any nonlinear transformation in their CR automorphism groups. In the recent work [18], the author proved the maximum conjecture in CR dimension one by applying the celebrated classical approach ofÉlie Cartan toward solving (biholomorphic) equivalence problems. Moreover, Gammel and Kossovskiy in [7] confirmed it in length ρ = 3.…”

Totally Nondegenerate Models and Standard Manifolds in CR Dimension One

“…Definition 1.1. (see [12,Definition 1.1]). An arbitrary (local) real analytic CR generic submanifold M ⊂ C 1+k of CR dimension one and codimension k is totally nondegenerate of the length ρ whenever the distribution D 1 = T 1,0 M + T 0,1 M is regular with the minimum possible degree of nonholonomy ρ.…”

“…Thus, as a frame for the complexified bundle C ⊗ T M , the distribution D ρ is generated by the iterated brackets between L and its conjugation L up to the length ρ. Following [12], let us show this frame by:…”