A classical idea going back at least to work of Leon Simon (1997) is that Liouville theorems for solutions to elliptic or parabolic PDEs are equivalent to Schauder-type regularity estimates. In this talk I will survey some recent developments of this idea concerning the regularity of the complex Monge-Ampère equation with respect to singular or degenerating reference Kähler metrics. The talk is partly based on joint work with V. Tosatti and M.-C. Lee (on collapsing Calabi-Yau fibrations and Kähler-Ricci flows) and work of my former student J. Klemmensen (on Calabi-Yau metrics with isolated conical singularities). This talk is part of the activities of the FIS2 project BeyondCY3 - CUP B53C25001070001. More information available at: https://sites.google.com/uniroma1.it/lfoscolo/fis2-beyondcy3/events