  We derive and analyse an energy to model lipid raft formation on biological
membranes involving a coupling between the local mean curvature and the local
composition. We apply a perturbation method recently introduced by Fritz, Hobbs
and the first author to describe the geometry of the surface as a graph over an
undeformed Helfrich energy minimising surface. The result is a surface
Cahn-Hilliard functional coupled with a small deformation energy We show that
suitable minimisers of this energy exist and consider a gradient flow with
conserved Allen-Cahn dynamics, for which existence and uniqueness results are
proven. Finally, numerical simulations show that for the long time behaviour
raft-like structures can emerge and stablise, and their parameter dependence is
further explored.
