A proof of the Willmore conjecture

Abstract

A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all spectral curves corresponding to periodic solutions of the Davey-Stewartson equation. The subsets of this moduli space, which correspond to bounded first integrals, are shown to be compact. With respect to another topology the moduli space is shown to be a Banach manifold. The subset of all periodic solutions of the Davey-Stewartson equation, which correspond to immersion of tori into the three-dimensional Euclidean space, are characterized by a singularity condition on the corresponding spectral curves. This yields a proof of the existence of minimizers for all conformal classes and the determination of the absolute minimum, which is realized by the Clifford torus.

215 pages

References in corpus (2)

• Quaternionic holomorphic geometry: Pluecker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori
• Conformal invariant functionals of immersions of tori into R^3

Cited by in corpus (14)

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• Families of conformal tori of revolution in the 3-sphere
• Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space
• The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere
• Constrained Willmore Tori in the 4--Sphere
• Discrete holomorphic geometry I. Darboux transformations and spectral curves
• On closed finite gap curves in spaceforms II
• On Closed Finite Gap Curves in Spaceforms I
• Harmonic tori in de Sitter spaces $S_1^{2n}$
• Willmore tori in the 4-sphere with nontrivial normal bundle