A proof of the Willmore conjecture
A proof of the Willmore conjecture
复制标题
威尔莫尔猜想的证明
DOI：
--
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
M. Schmidt
中科院分区：
文献类型：
--
作者：
M. Schmidt
给出了Willmore猜想的一个证明。借助整体Weierstrass表示，将Willmore泛函的变分问题转化为Davey-Stewartson方程周期解所对应的所有谱曲线的模空间上的约束变分问题.这个模空间的子集，对应于有界的第一积分，被证明是紧凑的。相对于另一个拓扑的模空间被证明是一个Banach流形。Davey-Stewartson方程的所有周期解的子集对应于环面浸入三维欧氏空间，其特征在于对应的谱曲线上的奇异性条件。这产生了一个证明的存在性极小的所有共形类和确定的绝对最小值，这是实现的克利福德环面。
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.