Willmore conjecture
On February 27, 2012, Fernando Codá Marques and André Neves announced a complete proof of the Willmore conjecture on a preprint posted to arxiv.org, using the min-max theory of minimal surfaces.
下面是我的笔记. (Here is my notes on this article in Chinese.)
Title: Min-Max Theory and the Willmore Conjecture
Author(s): Fernando C. Marques and André Neves
摘要
1965年, T. J. Willmore 给出了一个猜想, 对于浸入 $\mathbb{R}^3$ 中的环面, 其平均曲率的平方在环面上的积分至少为 $2\pi^2$. 我们使用极小曲面的 min-max 理论来证明这个猜想.
目录
1. 介绍
2. 主要思想及论文的组织
Part I. Willmore 猜想的证明
3. Canonical family: First properties
4. Definitions from Geometric Measure Theory
5. Canonical family:Boundary blow-up
6. Min-max family
7. Almgren-Pitts Min-Max Theory I
8. Almgren-Pitts Min-Max Theory II
9. Lower bound on width
10. 定理 B 的证明
11. 定理 A 的证明
Part II. Technical work
12. No area concentration
13. Interpolation results: Continuous to discrete
14. Interpolation results: Discrete to Continuous
15. Pull-tight
Appendix A.
Appendix B.
Appendix C.
References