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

We extend Tsuji's iterative construction of complete K\"ahler--Einstein metrics with negative scalar curvature to noncompact K\"ahler manifolds with bounded geometry, using Berndtsson's method from 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, and where the smooth fibers admit K\"ahler--Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise K\"ahler--Einstein metrics induce a semipositively curved metric on the relative canonical bundle $K_{X/Y}$. Moreover, our approach also applies to the plurisubharmonic variation of cusp K\"ahler--Einstein metrics.