  Liquid crystals with molecules constrained to the tangent bundle of a curved
surface show interesting phenomena resulting from the tight coupling of the
elastic and bulk free energies of the liquid crystal with geometric properties
of the surface. We derive thermodynamically consistent Frank-Oseen-Helfrich and
Landau-de Gennes-Helfrich models which consider the simultaneous relaxation of
the director/Q-tensor fields and the surface. The resulting systems of vector-
or tensor-valued surface partial differential equation and geometric evolution
laws are numerically solved to tackle the rich dynamics of these systems and to
compute the resulting equilibrium shapes. The results strongly depend on the
intrinsic and extrinsic curvature contributions and can lead to unexpected
asymmetric shapes.
