Parabolic Monge-Amp\`ere equations on a family of Calabi-Yau manifolds

Abstract

We prove a uniform $L^\infty$-estimate for the parabolic Monge--Amp\`ere equation under uniform H\"ormander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled K\"ahler--Ricci flows on a family of polarized Calabi--Yau manifolds over the punctured disk. An interpolation argument gives uniform H\"ormander-Tian type 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.