Willmore energy

In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere.

Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature.

is the Gaussian curvature, and dA is the area form of S. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic

The Willmore energy can be considered a functional on the space of embeddings of a given surface, in the sense of the calculus of variations, and one can vary the embedding of a surface, while leaving it topologically unaltered.

A basic problem in the calculus of variations is to find the critical points and minima of a functional.

For a given topological space, this is equivalent to finding the critical points of the function since the Euler characteristic is constant.

For embeddings of the sphere in 3-space, the critical points have been classified:[1] they are all conformal transforms of minimal surfaces, the round sphere is the minimum, and all other critical values are integers greater than 4

This flow leads to an evolution problem in differential geometry: the surface

is evolving in time to follow variations of steepest descent of the energy.

Like surface diffusion it is a fourth-order flow, since the variation of the energy contains fourth derivatives.