On Chern-Moser normal forms of strongly pseudoconvex hypersurfaces with high-dimensional stability group

TL;DR

This paper explicitly classifies strongly pseudoconvex non-spherical real-analytic hypersurfaces in complex spaces with high-dimensional stability groups, using Chern-Moser normal forms, extending previous classifications for certain automorphism group dimensions.

Contribution

It provides explicit descriptions of hypersurfaces with specific high-dimensional automorphism groups, filling gaps in the classification of such geometric structures.

Findings

Classified hypersurfaces with automorphism group dimension n^2-2n+1 for n≥2

Classified hypersurfaces with automorphism group dimension n^2-2n for n≥3

Extended previous classifications to new high-dimensional cases

Abstract

We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either $n^2-2n+1$ with $n\ge 2$, or $n^2-2n$ with $n\ge 3$. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi non-degenerate hypersurfaces in $\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint work by V. Ezhov and the author for the case $d_0(M)\ge n^2-2n+2$.