On the variation of curvature functionals in a space form 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 of constant sectional curvature. We calculate the first and second variations of this functional, extending known results and providing computationally accessible expressions given entirely in terms of the basic geometric information found in the surface fundamental forms. Further, we motivate and introduce the p-Willmore energy functional, applying the stability criteria afforded by our calculations to prove a result about the p-Willmore energy of spheres. 1 2 ON THE VARIATION OF CURVATURE FUNCTIONALSHelfrich model for membrane energy per unit area is given by the functionalwhere H is the membrane mean curvature, K is its Gauss curvature, k, k c are some rigidity constants, and c 0 is a constant known as the "spontaneous curvature". Physically, this high dependence on curvature arises from hydrostatic pressure differences between the fluids internal and external to the membrane.Another noteworthy curvature functional is the bending energy, which quantifies how much (on average) a surface M deviates from being a round sphere. Specifically, the bending energy functional is defined aswhere k 0 is the sectional curvature of the ambient space. This type of energy was first considered by Sophie Germain in 1811 (see [12]) as a model for the bending energy of a thin plate. In particular, she suggested that the bending energy be measured by an integral over the plate surface, taking as integrand some symmetric and even-degree polynomial in the principal curvatures. Note that the functional (2) is one of the simplest models of this kind.Remark. The bending energy also arises in the field of computer vision, where changes in surface curvature are used to simulate natural movement. On the other hand, it is known to these scientists as the surface torsion (see [13]) due to how it measures the change in normal curvature of the surface.From a mathematical perspective, both the Helfrich energy and the bending energy are closely related to the conformally invariant (see [14]) Willmore energy popularized in [15], which is defined as(3) W(M ) :=

Generation of Tubular and Membranous Shape Textures with Curvature Functionals

“…However, one may want to construct a general curvature functional where p is a smooth function with no symmetry constraint. A similar 2 form is given in [43,106], for a smooth function q(H , K ). A generalization to functions that depend on the position and the normal to the surface can be found in [19,26].…”

“…In contrast to these models, the polynomial p is here not required to be symmetric in the principal curvatures, which allows the generation of tubules. Other generalizations have been proposed in [43,106] and [19,26].…”

Critical tori for mean curvature energies in Killing submersions

“…Finally, from the definition of A, we have that A(η) = 4H 2 φ − 2K S + 2R η, and then the Euler-Lagrange equation boils down to [18] when the energy does not depend on the Gaussian curvature. In particular, let P (2H φ ) = H 2 φ and ρ = 0, i.e.…”

On p-Willmore Disks with Boundary Energies

“…By considering an arbitrary surface energy density depending on the mean curvature, we obtain an extension of the first term in the Helfrich energy H[X], (3), which includes (but is not restricted to) the original suggestions of S. Germain [11] as well as the Willmore and p-Willmore energies (see e.g. [14]). The additional inclusion of a total Gaussian curvature term is motivated mathematically by the conformal Willmore energy W[X], (2), and the Helfrich energy H[X], (3), but also plays an important role from a physical viewpoint.…”

“…Section 4 restricts to the special case of energy densities of the type P (H) = σ(H − c o ) p for σ > 0, c o ∈ R and non-negative integer p, i.e. p-Willmore energies with spontaneous curvature [14]. Equilibria are studied for each possible choice of the parameter p, paying special attention to the "ground state" H ≡ H o = c o which appears when p ≥ 2.…”