Min-max theory and the Willmore conjecture. (English) Zbl 1297.49079

The Willmore energy of a compact immersed surface in Euclidean space is the integral of the square of the mean curvature. In a short paper, T. J. Willmore [An. Sti. Univ. Al. I. Cuza Iasi, N. Ser., Sect. Ia 11B, 493–496 (1965; Zbl 0171.20001)] initiated the study of that functional by considering its value on compact surfaces smoothly embedded in \(3\)-dimensional Euclidean space. Willmore required but a few lines to prove that \({\mathcal W}(\Sigma) \geq 4\pi\), with equality if and only if \(\Sigma\) is a round sphere. He also calculated the value of \({\mathcal W}(\Sigma)\) for a torus of the form \[ (u,v) \mapsto \Big( (a+ b \cos u) \cos v,\, (a + b \cos u)\sin v,\, b \sin v\Big) \,, \] for which the minimum value, \({\mathcal W}(\Sigma) = 2\pi^2\), occurs when \(a/b =\sqrt{2}\). Willmore then conjectured that \({\mathcal W}(\Sigma)\geq 2\pi^2\) holds for every torus. This simply stated conjecture has been quite fruitful, leading to many significant developments. The paper under review culminates this work by affirmatively resolving the Willmore conjecture.
By means of the stereographic projection from the punctured 3-sphere onto the 3-dimensional Euclidean space the Willmore conjecture can be reformulated as a statement concerning a torus in the 3-sphere. The conjecture is then that the minimum of the Willmore energy is realized by the Clifford torus. In the paper under review, the authors prove specifically that, in the 3-sphere, any embedded closed surface of genus at least one has Willmore energy at least as large as that of the Clifford torus and equality holds if and only if the surface is conformally equivalent to the Clifford torus. L. Simon [Commun. Anal. Geom. 1, No. 2, 281–326 (1993; Zbl 0848.58012)] proved that in the \(n\)-dimensional Euclidean space there exists a compactly embedded real analytic torus that realizes the minimum of the Willmore energy, and a work of P. Li and S.-T. Yau [Invent. Math. 69, 269–291 (1982; Zbl 0503.53042)] shows that if a torus in the 3-sphere is not embedded, then it will have Willmore energy strictly larger than that of the Clifford torus. Thus the authors’ result proves the Willmore conjecture.
To obtain their result, the authors apply min-max theory in the context of geometric measure theory. This approach was introduced by F. J. Almgren, Jr. in mimeographed notes that first circulated in 1965, but which remain unpublished. J. T. Pitts [Existence and regularity of minimal surfaces on Riemannian manifolds. Princeton, New Jersey: Princeton University Press; University of Tokyo Press (1981; Zbl 0462.58003)] built on and strengthened Almgren’s min-max method. Though there are other treatments of min-max methods, the authors use the Almgren-Pitts approach because it imposes weaker regularity and convergence conditions on the min-max family of surfaces.
The authors provide a substantial outline of their proof in the second section of the paper, temporarily putting aside technical difficulties while emphasizing the main ideas. Ultimately, the technical effort required by the authors to accomplish their proof is substantial.


49Q10 Optimization of shapes other than minimal surfaces
49Q20 Variational problems in a geometric measure-theoretic setting
49J35 Existence of solutions for minimax problems
53A30 Conformal differential geometry (MSC2010)
53A05 Surfaces in Euclidean and related spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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Decimal expansion of 2*Pi^2.


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