IUMJ

Title: Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces

Authors: Jianguo Cao and Shu-Cheng Chang

Issue: Volume 56 (2007), Issue 6, 2839-2858

Abstract:

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.