Complete normal forms for real hypersurfaces in $\mathbb C^3$ at 2-nondegenerate points of Levi non-uniform rank zero

Abstract

We construct complete normal forms for $5$-dimensional real hypersurfaces in $\mathbb C^3$ which are $2$-nondegenerate and also of Levi non-uniform rank zero at the origin point ${\bf p} =0$. The latter condition means that the rank of the Levi form vanishes at ${\bf p}$ but not identically in a neighborhood of it. The mentioned hypersurfaces are the only finitely nondegenerate real hypersurfaces in $\mathbb C^3$ for which their complete normal forms were absent in the literature. As a byproduct, we also treat the underlying biholomorphic equivalence problem between the hypersurfaces. Our primary approach in constructing the desired complete normal forms is to utilize the techniques derived in the theory of equivariant moving frames. It notably offers the advantage of systematic and symbolic manipulation of the associated computations.