Willmore energy

Expressed symbolically, the Willmore energy of S is:

${\displaystyle {\mathcal {W}}=\int _{S}H^{2}\,dA-\int _{S}K\,dA}$

where ${\displaystyle H}$ is the mean curvature, ${\displaystyle K}$ 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 ${\displaystyle \chi (S)}$ of the surface, so

${\displaystyle \int _{S}K\,dA=2\pi \chi (S),}$

which is a topological invariant and thus independent of the particular embedding in ${\displaystyle \mathbb {R} ^{3}}$ that was chosen. Thus the Willmore energy can be expressed as

${\displaystyle {\mathcal {W}}=\int _{S}H^{2}\,dA-2\pi \chi (S)}$

An alternative, but equivalent, formula is

${\displaystyle {\mathcal {W}}={1 \over 4}\int _{S}(k_{1}-k_{2})^{2}\,dA}$

where ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$ are the principal curvatures of the surface.

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

${\displaystyle \int _{S}H^{2}\,dA}$

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.

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${\displaystyle \pi }$. They are called Willmore surfaces.

The Willmore flow is the geometric flow corresponding to the Willmore energy; it is an ${\displaystyle L^{2}}$-gradient flow.

${\displaystyle e[{\mathcal {M}}]={\frac {1}{2}}\int _{\mathcal {M}}H^{2}\,\mathrm {d} A}$

where H stands for the mean curvature of the manifold ${\displaystyle {\mathcal {M}}}$.

Flow lines satisfy the differential equation:

${\displaystyle \partial _{t}x(t)=-\nabla {\mathcal {W}}[x(t)]\,}$

where ${\displaystyle x}$ is a point belonging to the surface.

This flow leads to an evolution problem in differential geometry: the surface ${\displaystyle {\mathcal {M}}}$ 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.

• Cell membranes tend to position themselves so as to minimize Willmore energy.[2]
• Willmore energy is used in constructing a class of optimal sphere eversions, the minimax eversions.

• Willmore conjecture

• Willmore, T. J. (1992), "A survey on Willmore immersions", Geometry and Topology of Submanifolds, IV (Leuven, 1991), River Edge, NJ: World Scientific, pp. 11–16, MR 1185712.