Willmore energy

Summary

The Willmore energy of a closed surface in space is the integral of the square of its mean curvature over the surface. It measures how far the surface is from being round: it is at least $4\pi$, with equality only for round spheres. It does not change under scaling, and more generally under Möbius transformations of space. It appears in the study of elastic membranes, such as cell membranes, where bending costs energy.