Variations of Kähler–Einstein metrics on strongly pseudoconvex domains

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Let \((z,s)\in \mathbb {C}^n\times \mathbb {C}^m\) and \(\pi :\mathbb {C}^n\times \mathbb {C}^m\rightarrow \mathbb {C}^m\) be the projection on the second factor. Let \(D\) be a smooth domain in \(\mathbb {C}^{n+m}\) such that for each \(s\in \pi (D)\), the \(n\)-dimensional slice \(D_s=D\cap \pi ^{-1}(s)=\{z:(z,s)\in {D}\}\) is a smooth bounded strongly pseudoconvex domain. By a theorem of Cheng and Yau, for each slice \(D_s\) there exists a unique complete Kahler–Einstein metric \(h_{\alpha \bar{\beta }}(z,s)\) with a negative constant Ricci curvature. We prove that if the slice dimension \(n\) is greater than or equal to \(3\), then \(\log \det (h_{\alpha \bar{\beta }})_{1\le \alpha ,\beta \le {n}}\) is a plurisubharmonic function on \(D\). We also prove that it is a strictly plurisubharmonic function if \(D\) is a strongly pseudoconvex domain.