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. It is named after the English geometer Thomas Willmore.
Expressed symbolically, the Willmore energy of S is:
l{W}=\intSH2dA-\intSKdA
where
H
K
\chi(S)
\intSKdA=2\pi\chi(S),
which is a topological invariant and thus independent of the particular embedding in
R3
l{W}=\intSH2dA-2\pi\chi(S)
An alternative, but equivalent, formula is
l{W}={1\over4}\intS(k1-
| 2 | |
| k | |
| 2) |
dA
where
k1
k2
The Willmore energy is always greater than or equal to zero. A round sphere has zero Willmore energy.
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
\intSH2dA
One can find (local) minima for the Willmore energy by gradient descent, which in this context is called Willmore flow.
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
\pi
The Willmore flow is the geometric flow corresponding to the Willmore energy;it is an
L2
e[{
|
\intl{M
l{M}
Flow lines satisfy the differential equation:
\partialtx(t)=-\nablal{W}[x(t)]
x
This flow leads to an evolution problem in differential geometry: the surface
l{M}