# Infinitesimal symmetries of weakly pseudoconvex manifolds

## TL;DR

This paper classifies the infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces in complex space, revealing structural restrictions and jet determination properties crucial for understanding local equivalence.

## Contribution

It provides a complete classification of Lie algebras of infinitesimal CR automorphisms for weakly pseudoconvex hypersurfaces, including nonexistence of certain automorphisms and jet determination results.

## Key findings

- No nonlinear rigid automorphisms exist
- CR automorphisms are determined by 1-jets for non-balanced hypersurfaces
- Lie algebra of automorphisms is three graded for homogeneous polynomials

## Abstract

We classify the Lie algebras of infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces of finite multitype in $\mathbb C^N$. In particular, we prove that such manifolds admit neither nonlinear rigid automorphisms, nor real or nilpotent rotations. As a consequence, this leads to a proof of a sharp 2-jet determination result for local automorphisms. Moreover, for hypersurfaces which are not balanced, CR automorphisms are uniquely determined by their 1-jets. The same classification is derived also for special models, given by sums of squares of polynomials. In particular, in the case of homogeneous polynomials the Lie algebra of infinitesimal CR automorphisms is always three graded. The results provide an important necessary step for solving the local equivalence problem on weakly pseudoconvex manifolds.