Spatiotemporally coordinated transformations in epithelial curvature are necessary to generate crypt-villus structures during intestinal development. However, the temporal regulation of mechanotransduction pathways that drive crypt morphogenesis remains understudied. Intestinal organoids have proven useful to study crypt morphogenesis in vitro, yet the reliance on static culture scaffolds limits the ability to assess the temporal effects of changing curvature. Here, a photoinduced hydrogel cross-link exchange reaction is used to spatiotemporally alter epithelial curvature and study how dynamic changes in curvature influence mechanotransduction pathways to instruct crypt morphogenesis. Photopatterned curvature increased membrane tension and depolarization, which was required for subsequent nuclear localization of yes-associated protein 1 (YAP) observed 24 hours following curvature change. Curvature-directed crypt morphogenesis only occurred following a delay in the induction of differentiation that coincided with the delay in spatially restricted YAP localization, indicating that dynamic changes in curvature initiate epithelial curvature-dependent mechanotransduction pathways that temporally regulate crypt morphogenesis.