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.
Computer-Assisted Proof of the Main Theorem of 'The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere'
我们使用免费的数学软件Sage对论文“The Classification of Branched Willmore spheres in the $3$ -Sphere and the $4$ -Sphere”（arXiv：1706.01405）中的主要定理4.11中的四次和八次亚纯微分的全纯性进行了计算机辅助证明。
We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in normal Banach manifolds equipped with complete Finsler structures. We call the resulting tree of minmax problems a minmax hierarchy. We give several examples and in particular we explain how to implement this scheme in the framework of the viscosity method introduced by the author some years ago in order to give a new proof of the Willmore conjecture after the famous result by Marques and Neves.
We study the geometric significance of Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain in an odd-dimensional Euclidean space, we show that the asymptotic expansion of the magnitude function at infinity determines the Willmore energy of the boundary. This disproves the Leinster-Willerton conjecture for a compact convex body in all odd dimensions.
Uniqueness of equipotential photon surfaces in 4-dimensional static vacuum asymptotically flat spacetimes for positive, negative, and zero mass -- and a new partial proof of the Willmore inequality
The Willmore Problem seeks the surface in $\mathbb S^3\subset\mathbb R^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |\mathbf{H}_{\mathbb{R}^4}|^2 = \operatorname{area} + \int H_{\mathbb{S}^3}^2$. The longstanding Willmore Conjecture that the Clifford torus minimizes $W$ among genus-$1$ surfaces is now a theorem of Marques and Neves [19], but the general conjecture [10] that Lawson's [16] minimal surface $ξ_{g,1}\subset\mathbb S^3$ minimizes $W$ among surfaces of genus $g>1$ remains open. Here we prove this conjecture under the additional assumption that…
A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8π$. Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution …
