Euclidean Complete Hypersurfaces with Negative Constant Affine Mean Curvature

Given a constant L1 < 0 and a bounded convex domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb{R}}^{n}$$\end{document} with prescribed boundary value data in terms of a smooth strictly convex function φ, we construct an Euclidean complete hypersurface with constant affine mean curvature L1.