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