  Adsorption of 4He on atomically flat substrates such as graphene provides a
route towards the engineering of low dimensional quantum phases including
superfluids and strongly interacting insulators. In this study, we explore the
effects of graphene corrugation on the helium adsorption process via quantum
Monte Carlo simulations in the grand canonical ensemble. We utilize an
empirical adsorption potential based on the superposition of individual
helium-carbon interactions and systematically control corrugation, from a
smooth membrane to the fully rough potential, via the implementation of a
cutoff in reciprocal space. The results highlight the importance of using a
fully corrugated potential to understand the plethora of commensurate and
incommensurate phases in the first adsorbed layer. Surprisingly, some residual
effects of corrugation are still present before and during the promotion and
onset of a second adsorbed layer, where a smooth adsorption can lead to
enhanced particle fluctuations in a helium-interaction dominated regime.
