On the variation of curvature functionals in space forms with application to a generalized Willmore energy

Abstract

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.