LewyHypersurface 2004-11-16 11:47:51 2004-11-16 11:47:51 Example hypersurface % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The real hypersurface in $(z_1,\ldots,z_n) \in {\mathbb{C}}^n$ given by \begin{equation*} \operatorname{Im} z_n = \sum_{j=1}^{n-1} \lvert z_j \rvert^2 \end{equation*} is called the {\em Lewy hypersurface}. Note that this is a real hypersurface of real dimension $2n-1$. 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. \begin{thebibliography}{9} \bibitem{ber:submanifold} M.\@ Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. {\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}}, Princeton University Press, Princeton, New Jersey, 1999. \end{thebibliography}