Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

We prove a uniform $L^{\mathrm{\infty }}$-estimate for the parabolic Monge–Ampère equation under uniform Skoda estimates with respect to the initial and prescribed measures. We apply this estimate to Kähler–Ricci flows on a family of polarized Calabi–Yau manifolds over the punctured disk. An interpolation argument gives uniform Skoda estimates along the flows. We also establish uniform diameter bounds and Gromov–Hausdorff precompactness for this family of flows. In particular, we obtain an alternative proof of the diameter estimate by Li-Tosatti.