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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Lewy's curves and chains on real hypersurfaces


Author: James J. Faran
Journal: Trans. Amer. Math. Soc. 265 (1981), 97-109
MSC: Primary 32C05; Secondary 32F25
DOI: https://doi.org/10.1090/S0002-9947-1981-0607109-0
MathSciNet review: 607109
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Abstract: Lewy's curves on an analytic real hypersurface $ M = \{ r(z,z) = 0\} $ in $ {{\mathbf{C}}^2}$ are the intersections of $ M$ with any of the Segre hypersurfaces $ {Q_w} = \{ z:r(z,w) = 0\} $. If $ M$ is the standard unit sphere, these curves are chains in the sense of Chern and Moser. This paper shows the converse in the strictly pseudoconvex case: If all of Lewy's curves are chains, $ M$ is locally biholomorphically equivalent to the sphere. This is proven by analyzing the holomorphic structure of the space of chains. A similar statement is true about real hypersurfaces in $ {{\mathbf{C}}^n}$, $ n > 2$, in which case the proof relies on a pseudoconformal analogue to the theorem in Riemannian geometry which states that a manifold having "sufficiently many" totally geodesic submanifolds is projectively flat.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0607109-0
Article copyright: © Copyright 1981 American Mathematical Society