Willmore energy

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The Willmore energy of a surface is the integral of the mean curvature (minus two pi times the Euler characteristic). Cell membranes tend to position themselves so as to minimize Willmore energy. It is named after the English geometer Tom Willmore.

Expressed symbolically it is

$\mathcal{W} = \int_S H^2 \, dS - \int_S K \, dS \qquad \qquad (1)$

where S is the surface whose Willmore energy is to be computed, $H$ is the mean curvature, and $K$ is the Gaussian curvature. Note that the integral of the Gaussian curvature is proportional to the Euler characteristic

$\int_S K \, dS = 2 \pi \chi$ ,

which is a topological invariant.

Willmore energy is positive-definite, meaning that it is always greater or equal to zero. A sphere has zero Willmore energy.

An alternative formula equivalent to equation (1) is

$\mathcal{W} = {1 \over 4} \int_S (k_1 - k_2)^2 \, dS \qquad \qquad (2)$

where $k 1$ and $k 2$ are the principal curvatures of the surface.

For certain computing applications it may be desirable to vary the positional shape of a surface (while leaving it topologically unaltered) so as to minimize the Willmore energy of that surface. In such case, since the surface is left topologically unaltered, its Euler characteristic remains constant, so the Willmore energy is simplified to

$\mathcal{W} = \int_S H^2 \, dS$ .