On the maximum conjecture

TL;DR

This paper verifies the maximum conjecture on the rigidity of certain CR manifolds in specific cases, using Tanaka theory, and finds models satisfying the conjecture across various dimensions and lengths.

Contribution

It confirms the maximum conjecture for all CR dimension one models and full-models with free CR symbol algebras, expanding understanding of CR manifold rigidity.

Findings

Verification of the maximum conjecture for CR dimension one models

Confirmation for full-models with free CR symbol algebras

Existence of models satisfying the conjecture in all dimensions and lengths >= 3

Abstract

We verify the maximum conjecture on the rigidity of totally nondegenerate model CR manifolds in the following two cases: (i) for all models of CR dimension one (ii) for the so-called full-models, namely those in which their associated symbol algebras are free CR. In particular, we discover that in each arbitrary CR dimension and length >= 3, there exists at least one totally nondegenerate model, enjoying this conjecture. Our proofs rely upon some recent results in the Tanaka theory of transitive prolongation of fundamental algebras.