Willmore conjecture

Abstract: so the two functional differ by a constant. The functional i^(X) has the advantage that its integrand is nonnegative and vanishes exactly at the umbilic points of the immersion X. Obviously iT(X) = 0 iff M = S and X is totally umbilic. Thus, the absolute minimum of #\"on the space of immersions X: S -> E 3 is 0 and the critical locus of such X is known. When M is a torus, Willmore provided an example of an immersion X: M -* E with iΓ(X) = 2ττ and showed that iΓ(X) > 2π for all smooth surfaces of revolution. He then conjectured that iΓ(X)^ 2*π for all immersions of the torus with equality only for the example he provided: the anchor ring swept out by revolving a circle of radius r about the line whose distance from the center of the circle was r]/ϊ. White then pointed out that the two-form (H K) dA had the property of being invariant under conformal transformations of E 3 plus the \"point at infinity\

Abstract: In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2\pi^2. We prove this conjecture using the min-max theory of minimal surfaces.

Abstract: Let 0: M-,S" be a minimal immersion of a compact surface into a unit sphere. Then, the linear functions of 0 are eigenfunctions for the Laplacian of M corresponding to the eigenvalue 2=2 . The main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non-zero eigenvalue of its Laplacian. This kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems, as Li and Yau have shown in [6]. The methods that we use in this paper are based, for the most part, on [6]. It is known that the only metric on a 2-dimensional sphere admitting a minimal immersion into S" by the first eigenfunctions is the standard one (this follows, for example, from the fact that the multiplicity of the first eigenvalue for such a metric is at most three, see the Cheng work [3]). Our first result shows that it is possible to extend this property for an arbitrary compact surface, in the following way: