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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A two-sided H. Lewy extension phenomenon

Authors: L. R. Hunt and M. Kazlow
Journal: Proc. Amer. Math. Soc. 74 (1979), 95-100
MSC: Primary 32D10
MathSciNet review: 521879
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Abstract: Let M be a $ {C^\infty }$ real $ (n + k)$-dimensional CR-manifold in $ {{\mathbf{C}}^n}$. We are interested in finding conditions on M near a point $ p \in M$ which imply that all CR-function on M extend to holomorphic functions in some fixed neighborhood of p in $ {{\mathbf{C}}^n}$. Of course if M is a real hypersurface, it is known that M having eigenvalues of opposite sign in its Levi form at p will give us such an extension result. If we view the Levi form at a point on a general CR-manifold M as a quadratic map from the holomorphic tangent space to the normal space of the real tangent space in $ {{\mathbf{C}}^n}$, and if this map is surjective, then we prove our CR-functions extend to holomorphic functions in an open neighborhood of the point. We also show that if the real codimension of M in $ {{\mathbf{C}}^n}$ is 2, and if the Levi form is totally indefinite, then the Levi form is onto $ {{\mathbf{R}}^2}$ as a quadratic map, and hence we have our extension theory.

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Keywords: CR-functions, Levi form, quadratic maps, totally indefinite, CR-extension
Article copyright: © Copyright 1979 American Mathematical Society

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