  We consider sharp interface asymptotics for a phase field model of two phase
near spherical biomembranes involving a coupling between the local mean
curvature and the local composition proposed by the first and second authors.
The model is motivated by lipid raft formation. We introduce a reduced diffuse
interface energy depending only on the membrane composition and derive the
$\Gamma-$limit. We demonstrate that the Euler-Lagrange equations for the
limiting functional and the sharp interface energy coincide. Finally, we
consider a system of gradient flow equations with conserved Allen-Cahn dynamics
for the phase field model. Performing a formal asymptotic analysis we obtain a
system of gradient flow equations for the sharp interface energy coupling
geodesic curvature flow for the phase interface to a fourth order PDE free
boundary problem for the surface deformation.
