Explicit absolute parallelism for $2$-nondegenerate real hypersurfaces $M^5\subset {\mathbb{C}}^3$ of constant Levi rank $1$

Abstract

We study the local equivalence problem for five dimensional real hypersurfaces $M^5$ of ${\mathbb{C}}^3$ which are $2$-nondegenerate and of constant Levi rank $1$ under biholomorphisms. We find two invariants, $J$ and $W$, which are expressed explicitly in terms of the graphing function $F$ of $M$, the annulation of which give a necessary and sufficient condition for $M$ to be locally biholomorphic to a model hypersurface, the tube over the light cone. If one of the two invariants $J$ or $W$ does not vanish on $M$, we show that the equivalence problem under biholomophisms reduces to an equivalence problem between $\{e\}$-structures, that is we construct an absolute parallelism on $M$.

55 pages

References in corpus (1)

• Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

Cited by in corpus (3)

• Structure equations of Levi degenerate CR hypersurfaces of uniform type
• A complete normal form for everywhere Levi degenerate hypersurfaces in ${\mathbb{C}}^3$
• Fundamental invariants of 2--nondegenerate CR geometries with simple models