Kähler-Einstein fillings (Q2869828)

Let \(\Omega\) be a bounded smooth strongly pseudoconvex domain in \(\mathbb C^n\). The aim of this paper is to prove that there exists a smooth, up to the boundary, Kähler -Einstein metric on \(\Omega\) with positive Einstein constant such that the restriction of the metric to the Levi distribution of the boundary is conformal to the Levi form.NEWLINENEWLINEIt is a well known fact that the existence of such a metric can be reduced to finding a smooth strictly plurisubharmonic function \(u\) in \(\Omega\), a solution of the following complex Monge-Ampère equation NEWLINE\[NEWLINE (dd^cu)^n=\frac {e^{-u}d\lambda_{2n}}{\int_{\Omega}e^{-u}d\lambda_{2n}} \text{ in } \Omega, \;\;u=0 \text{ on } \partial \Omega, \tag{MAE} NEWLINE\]NEWLINE where \(\lambda_{2n}\) is normalized Lebesgue measure in \(\mathbb C^n\) such that \(\lambda_{2n}(\Omega)=1\).NEWLINENEWLINEFor a given smooth strictly plurisubharmonic function \(u_0\) in \(\Omega\) such that \(u_0=0\) on \(\partial \Omega\), let \(u_j\) be a sequence of smooth plurisubharmonic (unique) solutions of the following Monge-Ampère equations NEWLINE\[NEWLINE (dd^cu_{j})^n=\frac {e^{-u_{j-1}}d\lambda_{2n}}{\int_{\Omega}e^{-u_{j-1}}d\lambda_{2n}} \text{ in } \Omega, \;\;u_{j}=0 \text{ on } \partial \Omega, \, j=1,2,\dots NEWLINE\]NEWLINE The authors prove that the sequence \((u_j)\) is bounded in \(\mathcal C^{\infty}(\overline{\Omega})\) and has a subsequence convergent to a smooth plurisubharmonic function \(u\), a solution of (MAE).NEWLINENEWLINELater the uniqueness of the solution \(u\) is discussed in some spacial class of domains. Let \(\Omega\) be a bounded, circled, smooth strictly pseudoconvex domain and let \(u\) be a smooth \(S^1\)-invariant strictly plurisubharmonic solution of (MAE). If in addition \(\Omega\) is strictly \(u\)-convex, then \(u\) is the unique \(S^1\)-invariant solution. In particular, if \(\Omega=B(a,r)\) is a ball in \(\mathbb C^n\) of radius \(r\in (0,1)\), then there exists a unique \(S^1\)-invariant solution of (MAE) on \(B(a,z)\), moreover \(u\) is a radial function.