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Complex Dimension Real Number Sphere Linear Manifold Biholomorphically Equivalent Hypersurface

1	Dimension
1	Complex
1	Real Number
2	Sphere
4	Linear Manifold
8	Biholomorphically Equivalent
10	Hypersurface

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Lewy Hypersurface

The real hypersurface in $(z_1,\ldots,z_n) \in {\mathbb{C}}^n$ given by

$\displaystyle \operatorname{Im} z_n = \sum_{j=1}^{n-1} \lvert z_j \rvert^2$

is called the Lewy hypersurface. Note that this is a real hypersurface of real dimension

. This is an example of a non-trivial real hypersurface in complex space. For example it is not biholomorphically equivalent to the hyperplane defined by $\operatorname{Im} z_n = 0$ , but it is locally (not globally) biholomorphically equivalent to a unit sphere.

Bibliography

1: M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.