Duke Mathematical Journal

Uniformly Levi degenerate CR manifolds: The 5-dimensional case

Peter Ebenfelt

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Abstract

In this paper, we consider real hypersurfaces $M$ in $\mathbb {C}^3$ (or, more generally, $5$-dimensional CR (Cauchy-Riemann) manifolds of hypersurface type) at uniformly Levi degenerate points, that is, Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require that the hypersurface satisfy a certain second-order nondegeneracy condition (called $2$-nondegeneracy) at the point. One of our main results is the construction, near any point $p_0\in M$ satisfying the above conditions, of a principal bundle $P\to M$ and a $\mathbb {R}^{\dim P}$-valued 1-form $\underline {\omega}$, uniquely determined by the CR structure on $M$, which defines an absolute parallelism on $P$. If $M$ is real-analytic, then covariant derivatives of $\underline {\omega}$ yield a complete set of local biholomorphic invariants for $M$. This solves the biholomorphic equivalence problem for uniformly Levi degenerate hypersurfaces in $\mathbb {C}^3$ at $2$-nondegenerate points.

Article information

Source
Duke Math. J., Volume 110, Number 1 (2001), 37-80.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087574812

Digital Object Identifier
doi:10.1215/S0012-7094-01-11012-0

Mathematical Reviews number (MathSciNet)
MR1861088

Zentralblatt MATH identifier
1020.32029

Subjects
Primary: 32V05: CR structures, CR operators, and generalizations
Secondary: 32V20: Analysis on CR manifolds 32V40: Real submanifolds in complex manifolds

Citation

Ebenfelt, Peter. Uniformly Levi degenerate CR manifolds: The 5-dimensional case. Duke Math. J. 110 (2001), no. 1, 37--80. doi:10.1215/S0012-7094-01-11012-0. https://projecteuclid.org/euclid.dmj/1087574812


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