Characterization for balls by potential function of Kähler-Einstein metrics for domains in \(\mathbb C^n\) (Q812190)

It was shown by \textit{W. Stoll} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 7, 87--154 (1980; Zbl 0438.32005)] and \textit{D. Burns} [Ann. Math. (2) 115, 349--373 (1982; Zbl 0507.32011)] that complex manifolds biholomorphically equivalent to the unit ball \(B_n\) in \(\mathbb{C}^n\) can be characterized by existence of strictly parabolic exhaustion functions (that satisfy the homogeneous Monge-Ampère equation). In the present paper, smoothly bounded pseudoconvex domains \(D\subset\mathbb{C}^n\) that are biholomorphically equivalent to \(B_n\) are characterized in terms of plurisubharmonic exhaustions functions satisfying certain nonhomogeneous Monge-Ampère equations (potentials for the Kähler-Einstein metrics) as follows. Let a smooth plurisubharmonic function \(U\) satisfy \(\text{def\,}H(V)= e^{(n+1)V+h}\) in \(D\), \(U= \infty\) on \(\partial D\), \(\min\{U(z): z\in D\}= 0\), where \(H(V)\) is the complex Hesse matrix and \(h\) is a bounded pluriharmonic function in \(D\). If \[ \liminf_{z\to\partial D}\,\text{det\,}H(\log(1- e^{-U}))\geq 0, \] then \(D\) is biholomorphically equivalent to \(B_n\). If, in addition, \(h\equiv\text{const}\), then there exists a biholomorphic map \(\varphi: D\to B_n\) with constant Jacobian.