An iterative construction of complete K\"ahler--Einstein metrics

We extend Tsuji's iterative construction of a K\"ahler--Einstein metric with negative scalar curvature to (non-compact) K\"ahler manifolds with bounded geometry, using Berndtsson's method for the compact setting. Consequently, given a holomorphic surjective map $p:X \to Y$, where $X$ is a weakly pseudoconvex K\"ahler manifold and $Y$ is a complex manifold, whose smooth fibers admit K\"ahler--Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise K\"ahler--Einstein metric induces a semi-positively curved metric on the relative canonical bundle $K_{X/Y}$ of $p$. Moreover, our approach can be applied to obtain the plurisubharmonic variation of cusp K\"ahler--Einstein metrics.