Biholomorphic equivalence to totally nondegenerate model CR manifolds and Beloshapka's maximum conjecture

TL;DR

This paper uses Cartan's method to reduce the biholomorphic equivalence problem for certain CR manifolds to an absolute parallelism, proving Beloshapka's maximum conjecture about the rigidity of these models.

Contribution

It establishes a reduction of the equivalence problem to an absolute parallelism and proves the rigidity conjecture for CR models of length three or more.

Findings

Reduction of the equivalence problem to an absolute parallelism.

Proof of Beloshapka's maximum conjecture for models of length ≥ 3.

Rigidity of CR automorphism groups with no nonlinear maps.

Abstract

Applying Elie 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.