View source for I-beam

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 manifold|smooth]] [[closed surface]] [[Embedding|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]].

==Definition==
Expressed symbolically, the Willmore energy of ''S'' is:

:<math> \mathcal{W} = \int_S H^2 \, dA - \int_S K \, dA</math>

where <math>H</math> is the [[mean curvature]], <math>K</math> 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]] <math>\chi(S)</math> of the surface, so

:<math> \int_S K \, dA = 2 \pi \chi(S), </math>

which is a [[topological property|topological invariant]] and thus independent of the particular embedding in <math>\mathbb{R}^3</math> that was chosen.  Thus the Willmore energy can be expressed as
:<math> \mathcal{W} = \int_S H^2 \, dA - 2 \pi \chi(S)</math>

An alternative, but equivalent, formula is

:<math> \mathcal{W} = {1 \over 4} \int_S (k_1 - k_2)^2 \, dA</math>

where <math>k_1</math> and <math>k_2</math> are the [[principal curvatures]] of the surface.

===Properties===
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.

==Critical points==
A basic problem in the [[calculus of variations]] is to find the [[critical point (mathematics)|critical points]] and minima of a functional.

For a given topological space, this is equivalent to finding the critical points of the function
:<math>\int_S H^2 \, dA </math>
since the Euler characteristic is constant.

One can find (local) minima for the Willmore energy by [[gradient descent]], which in this context is called [[#Willmore_flow|Willmore flow]].

For embeddings of the sphere in 3-space, the critical points have been classified:<ref>Robert Bryant. A duality theorem for Willmore surfaces. J. Differential Geometry 20(1984), 23–53.</ref> they are all [[conformal transform]]s of [[minimal surface]]s, the round sphere is the minimum, and all other critical values are integers greater than or equal to 4<math>\pi</math>.

==Willmore flow==
The '''Willmore flow''' is the [[geometric flow]] corresponding to the Willmore energy;
it is an <math>L^2</math>-[[gradient flow]].

:<math>e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A</math>

where ''H'' stands for the [[mean curvature]] of the [[manifold]] <math>\mathcal{M}</math>.

Flow lines satisfy the differential equation:
:<math> \partial_t x(t) = -\nabla \mathcal{W}[x(t)] \, </math>
where <math>x</math> is a point belonging to the surface.

This flow leads to an evolution problem in [[differential geometry]]: the surface <math>\mathcal{M}</math> is evolving 
in time to follow variations of steepest descent of the energy. Like [[surface diffusion (mathematics)]] it is a fourth-order 
flow, since the variation of the energy contains fourth derivatives.

==Applications==
* [[Cell membrane]]s tend to position themselves so as to minimize Willmore energy.

* Willmore energy is used in constructing a class of optimal [[sphere eversion]]s, the [[minimax eversion]]s.

==See also==
* [[Willmore conjecture]]

==References==
<references/>
* Thomas J. Willmore. A survey on Willmore immersions. In Geometry and Topology of Submanifolds, IV (Leuven, 1991), pp 11–16. World Sci. Pub., 1992.

[[Category:Geometric flow]]
[[Category:Differential geometry]]
[[Category:Surfaces]]