  Functionals involving surface curvature are important across a range of
scientific disciplines, and their extrema are representative of physically
meaningful objects such as atomic lattices and biomembranes. Inspired in
particular by the relationship of the Willmore energy to lipid bilayers, we
consider a general functional depending on a surface and a symmetric
combination of its principal curvatures, provided the surface is immersed in a
3-D space form. We compute the first and second variations of this functional,
leading to expressions given entirely in terms of the surface fundamental
forms. We then apply the stability criteria afforded by our calculations to a
generalization of the Willmore functional, proving a result regarding the
stability of spheres.
