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EDIT, Monday, 21 March. I found my copy of Willmore's book, "Total Curvature in Riemannian Geometry." In sections 5.9, 5.10, pages 132-136, he gives all the material I had half-remembered. First, it is possible to have knotted tori in $R^3,$ this definitely increases the lower bound on the Willmore functional. With bridge number $n,$ the functional is at least $8 \pi n.$

The reason the conjectured optimum is called a Clifford torus is this: the problem is conformally invariant in the ambient space $R^3.$ Any minimal surface in $S^3$ maps to a stationary surface for the Willmore funtional by stereographic projection. The Clifford torus in the standard $S^3 \in R^4$ is $$x_1^2 + x_2^2 = x_3^2 + x_4^2 = \frac{1}{2}.$$If we pick the most fortunate point from which to project to an $R^3,$ we get a "round" torus. But if we rotate the surface first, then project, we get these funny bulbous tori, skinny on one side, fat on the other. So the conjecture should say "if and only if conformally equivalent to Willmore's anchor ring." Apparently on large nautical anchors, there is some use for a heavy iron ring in this shape, I don't know why, but that is a traditional name in English mathematics articles for a torus constructed by revolving a circle.

For variational problems, the first two steps in finding a minimizer are often finding a critical point or "stationary" point of the functional. Second is "stability," meaning, in a way, local minimum.

It is noteworthy that Lawson proved in his 1968 dissertation that there are closed minimal surfaces of arbitrary genus in $S^3.$

For those with a more differential geometric background, one ought not to ignore the influence of Robert L. Bryant in this field. In particular, 1984, Journal of Differential Geometry, "A Duality Theorem for Willmore Surfaces."

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Hi, there is just one viable candidate, a standard round torus with a certain ratio,$\sqrt 2,$ of the radius of revolution to the radius of the circle being revolved. Willmore himself proved that his candidate is optimal among some families of tori created by moving a circle of constant radius around a curve in $\mathbf R^3.$ Leon Simon at Stanford proved that there is a minimizer. My impression is that you are at Smith, I will check after I post this. The person you want to ask is Rob Kusner at U. Mass. Amherst. Rob has pointed out that the conjecture goes back to Blaschke and was rediscovered by Willmore. These things happen.

In 1982, Li and Yau introduced the conformal volume to deal with this problem. Their paper in Inventiones is available online.

Enough for now. A ton of stuff available on Giggle.