Question

It is my understanding that the torus that minimizes the Willmore energy is not yet known (this from a sentence in a 2005 paper by John Sullivan). Willmore conjectured that the Willmore energy for any smooth, immersed torus is $\ge 2 \pi^2$. From what I can gather, this energy is achieved by a standard rotationally round torus, derived from the Clifford torus.

Assuming the problem remains open, my question is:

Are there any viable candidates for different torus Willmore minimizers, or does the evidence point to the one just mentioned?

I'm wondering if the problem remains open because all strange, unlikely alternatives have not been ruled out, or because there are actually some viable alternatives. Likely the right reference I am not finding would suffice. Thanks!

Answer 1

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.

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."

Answer 2

Just a few days ago, Fernando C Marques and Andre Neves posted a preprint on the arxiv in which they (claim to) provide a complete proof of Willmore's Conjecture. I have no idea how much of it has been verified, especially given it is so recent, but the geometric ideas used (min-max theory for minimal submanifolds) seem very elegant.

Answer 3

Will Jagy already cited the Li and Yau reference about the conformal volumne. This can be used to prove the Willmore conjecture for a special class of conformal types for the tori, namely for those tori with conformal structure given by the lattice $\Z 1+\Z\tau$ in $\C,$ where the imaginary part of $\tau$ is smaller or equal to $1.$ This follows from the inequality $$\int_\Sigma H^2 dA\geq V_c(\Sigma)\geq\frac{\lambda}{2} vol(\Sigma,g_0),$$ where $V_c(\Sigma)$ is the conformal volumne, $g_0$ is the flat metric on the Riemann surface $\Sigma,$ and $\lambda$ is the first positive eigenvalue of the (flat) Laplacian of $(\Sigma,g_0).$ This can be computed explicitley, and one obtains $$\geq2\pi^2/y,$$ where $y=\Im(\tau).$

There are also other surfaces classes, for which the conjecture is proved, for example Kanaltori (see Pinkall & Jeromin-Hertrich).

There also exists an arxiv article of Martin Schmidt, who claims on the last page (of more then 200) that he has a proof of the Willmore conjecture. Even if the ideas in that have been quiet influential, it looks like there are at least some gaps in the proof.