Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces Jianguo CaoShu-Cheng Chang 32V1553C17partially-Einstein metricspseudo-Einstein metricsQ-curvatureCR-hypersurfacesKohn's $\bar{\partial}_b$ theory In this paper, we will use the Kohn $\bar{\partial}_b$-theory on CR-hypersurfaces to derive some new results in CR-geometry: Main Theorem. Let $M^{2n-1}$ be the smooth boundary of a bounded strongly pseudo-convex domain $\Omega$ in a complete Stein manifold $V^{2n}$. Then: \begin{enumerate}[{\upshape(1)}] \item For $n \ge 3$, $M^{2n-1}$ admits a pseudo-Einstein metric. \item For $n \ge 2$, $M^{2n-1}$ admits a Fefferman metric of zero CR $Q$-curvature. \item In addition, for a compact strictly pseudoconvex CR emendable 3-manifold $M^3$, its CR Paneitz operator $P$ is a closed operator. \end{enumerate} \noindent\upshape There are examples of non-emendable strongly pseudoconvex CR-manifolds $M^3$, for which the corresponding $\bar{\partial}_b$-operator and Paneitz operators are not closed operators. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3111 10.1512/iumj.2007.56.3111 en Indiana Univ. Math. J. 56 (2007) 2839 - 2858 state-of-the-art mathematics http://iumj.org/access/