On characterization for a class of pseudo-convex domains with positive constant pseudo-scalar curvature on their boundaries
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 \(M=\partial D\) and \(\theta ={1\over {2i}}(\partial \rho-\bar \partial \rho)\). Assume \(\log J(\rho)\) is harmonic in the Kähler metric \({{\partial^2u}\over {\partial z_i \partial \bar z_j}}dz_i\otimes d\bar z_j\). Then the following two statements hold: {\parindent=6mm\begin{itemize}\item[(a)] If \(R_\theta\equiv c>0\), constant on \(M \), then \(D\) is biholomorphic to the unit ball \(B^{n+1}\) in \({\mathbb C}^{n+1}\). \item[(b)] Webster pseudo-scalar curvature on \(M \) is given by \[ R_\theta=n(n+1){{\det H(\rho)}\over {J(\rho)}}. \] \end{itemize}}
scientific article; zbMATH DE number 5572864