  In this paper we introduce a mathematical model for small deformations
induced by external forces of closed surfaces that are minimisers of
Helfrich-type energies. Our model is suitable for the study of deformations of
cell membranes induced by the cytoskeleton. We describe the deformation of the
surface as a graph over the undeformed surface. A new Lagrangian and the
associated Euler-Lagrange equations for the height function of the graph are
derived. This is the natural generalisation of the well known linearisation in
the Monge gauge for initially flat surfaces. We discuss energy perturbations of
point constraints and point forces acting on the surface. We establish
existence and uniqueness results for weak solutions on spheres and on tori.
Algorithms for the computation of numerical solutions in the general setting
are provided. We present numerical examples which highlight the behaviour of
the surface deformations in different settings at the end of the paper.
