  Within a coupled-field Ginzburg-Landau model we study analytically phase
separation and accompanying shape deformation on a two-phase elastic membrane
in simple geometries such as cylinders, spheres and tori. Using an exact
periodic domain wall solution we solve for the shape and phase ordering field,
and estimate the degree of deformation of the membrane. The results are
pertinent to a preferential phase separation in regions of differing curvature
on a variety of vesicles.
