On characterization for a class of pseudo-convex domains with positive constant pseudo-scalar curvature on their boundaries
type: Item
instance of: scholarly article
title: On characterization for a class of pseudo-convex domains with positive constant pseudo-scalar curvature on their boundaries
zbMATH Open document ID: 1194.32021
author: Song-Ying Li
published in: Communications in Analysis and Geometry
publication date: 2009-06-30T00:00:00Z
review text: Let \(M\) be a compact strictly pseudoconvex real hypersurface in \({\mathbb C}^{n+1}\), and let \(\theta_0\) be a pseudo-Hermitian real 1-form on \(M\), defining a pseudo-Hermitian structure in the sense of \textit{S. M. Webster} [J. Differ. Geom. 13, 25--41 (1978; Zbl 0379.53016)]. By the solution of the CR Yamabe problem [see \textit{D. Jerison} and \textit{J. M. Lee}, J. Differ. Geom. 25, 167--197 (1987; Zbl 0661.32026); \textit{N. Gamara}, J. Eur. Math. Soc. 3, No. 2, 105--137 (2001; Zbl 0988.53013); \textit{N. Gamara} and \textit{R. Yacoub}, Pac. J. Math. 201, No. 1, 121--175 (2001; Zbl 1054.32020)], one can find \(\theta\), conformally equivalent to \(\theta_0\), with constant pseudo-scalar curvature~\(R_\theta\). Any pseudo-Hermitian real 1-form on \(M\) can be obtained as \(\theta_\rho={1\over {2i}}(\partial \rho-\bar \partial \rho)\), for some defining function \(\rho\) of \(M\). The purpose of this paper is to characterize those \(\rho\) for which one has {\parindent=6mm\begin{itemize}\item[(P)] \(R_{\theta_\rho}\) is equal to a positive constant if and only if \(M\) is CR equivalent to the unit sphere \(S^{2n+1}\). \end{itemize}} Recall that if \(D \subset {\mathbb C}^{n+1}\) is a smoothly bounded pseudo-convex domain, \(u\) is a strictly plurisubharmonic exhaustion function for \(D\) and \(\rho(z)=-e^{- u(z)}\), then the Fefferman functional \(J(\rho)\) of \(\rho\) is defined as \[ J(\rho)=-\det \begin{pmatrix} \rho&\bar\partial \rho\cr (\bar\partial \rho)^*& H(\rho)\end{pmatrix}, \] where \(H(\rho)\) is the complex Hessian of \(\rho\), \( \bar\partial \rho=(\partial_{\bar z_1}\rho,\dots,\partial_{\bar z_n}\rho)\) and \((\bar\partial \rho)^*\) is its adjoint. The main result of the paper is the following sufficient condition on \(\rho\) for (P) to hold: Theorem. Let \(D\subset \mathbb C^{n+1}\) be a domain and let \(\rho\in C^3(\overline D)\) be a defining function for \(D\) so that \(u(z)=-\log (-\rho(z))\) is strictly plurisubharmonic in \(D\). Let \(…
reviewed by: Laura Geatti
Mathematics Subject Classification ID: 32V20
zbMATH DE Number: 5572864
zbMATH Keywords: CR hypersurface
zbMATH Keywords: pseudo-hermitian structure
MaRDI profile type: Publication
DOI: 10.4310/CAG.2009.V17.N1.A2
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description: scientific article; zbMATH DE number 5572864