Willmore's Conjecture

Elastic bending energy or Willmore energy is a useful concept to people who mathematically analyse biological membranes. Some objects, such as red blood cells, may get their characteristic shapes by settling into configurations which minimise this energy, subject to other constraints. (Source: www.math.umd.edu/~rhn/papers/pdf/geom-biomembrane.pdf )

Willmore energy is geometrically interesting too. Technically, it is calculated by taking the mean curvature at each point on the surface, squaring it, and integrating the result over the whole shape. It has been known for a long time that the minimum energy which any topological sphere may have is 4π (which happens for the ordinary old sphere). But what of the next simplest type of surface, a torus?

In 1965, Tom Willmore conjectured that the minimal answer here is 2π², and last month Fernando Marques & André Neves uploaded a preprint claiming a proof: http://arxiv.org/abs/1202.6036

They show that this minimal value is achieved only by stereographic projections of conformal transformations of the 4-dimensional Clifford torus: http://en.wikipedia.org/wiki/Clifford_torus

All of which is to say that mathematicians have finally found the perfect doughnut: http://www.huffingtonpost.com/frank-morgan/math-finds-the-best-dough_b_1331844.html
[via +Moshe Vardi]
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