A C∞-Regularity Theorem for Nondegenerate CR Mappings

We prove the following regularity result: If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M\subset {\Bbb C}^N$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M^\prime\subset{\Bbb C}^{N^{\prime}}$\end{document} are smooth generic submanifolds and M is minimal, then every Ck-CR-map from M into M′ which is k-nondegenerate is smooth. As an application, every CR diffeomorphism of k-nondegenerate minimal submanifolds in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Bbb C}^N$\end{document} of class Ck is smooth.