Higher order symmetries of real hypersurfaces in $\Bbb C^3.$

TL;DR

This paper classifies real hypersurfaces in complex three-space with complex symmetry structures, revealing the role of higher order symmetries and providing insights into automorphism determination and longstanding conjectures.

Contribution

It offers a complete classification of Levi degenerate hypersurfaces of finite multitype in , analyzing higher order symmetries and confirming a conjecture about nonlinear automorphisms.

Findings

Higher order symmetries exist in Levi degenerate hypersurfaces in .

2-jets are insufficient to determine automorphisms in certain hypersurfaces.

The classification confirms a conjecture on the origin of nonlinear automorphisms.

Abstract

We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By recent results of Kolar, Meylan and Zaitsev, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in $\mathbb C^2$. We analyze this phenomenon for hypersurfaces of finite Catlin multitype in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on which hypersurfaces 2-jets are not sufficient to determine an automorphism. The results also confirm a conjecture about the origin of nonlinear automorphisms of Levi degenerate hypersurfaces, formulated by the first author (AIM 2010).