  Multi-component membranes such as ternary mixtures of lipids and cholesterol
can exhibit coexistence regions between two liquid phases. When such membranes
adhere to a corrugated substrate, the phase separation process strongly depends
on the interplay between substrate topography, bending rigidities, and line
tension of the membrane domains as we show theoretically via energy
minimization and Monte Carlo simulations. For sufficiently large bending
rigidity contrast between the two membrane phases, the corrugated substrate
truncates the phase separation process and leads to a stable pattern of
membrane domains. Our theory is consistent with recent experimental
observations and provides a possible control mechanism for domain patterns in
biological membranes.
