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.

References in corpus (8)

• Geometric theory on the elasticity of bio-membranes
• Nonrelativistic quantum dynamics on a cone with and without a constraining potential
• Total mean curvature, scalar curvature, and a variational analog of Brown-York mass
• A Gap Theorem for Willmore Tori and an application to the Willmore Flow
• On the total mean curvature of non-rigid surfaces
• Quantum-elastic bump on a surface
• A Conformal Field Theory of Extrinsic Geometry of 2-d Surfaces
• Topological Bounds of Bending Energy for Lipid Vesicles

Cited by in corpus (5)

• Computational p-Willmore Flow with Conformal Penalty
• Critical Tori for Mean Curvature Energies in Killing Submersions
• A relation between cylindrical critical points of Willmore-type energies, weighted areas and vertical potential energies
• Generation of tubular and membranous shape textures with curvature functionals
• On p-Willmore Disks with Boundary Energies