探討由最小-最大定理證明威爾莫猜想

A survey on proof of Willmore conjecture by min-max theory

劉書銘(Shu-Ming Liu)

指導教授 ： 張樹城

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.10349

摘要

2012年，Fernando C.Marques和Andre Neves所著的論文：Min-max theory and Willmore conjecture成功地證明了1965年由Willmore所提出著名的猜想。他們使用了幾何測度論的方法並且應用Min-max theory給出一個漂亮的證明方法，其中特別的是如何造出能夠使用Min-max theory的canonical family，這個是整篇論文中最重要的部分；我們將會在本文中說明如何造出這樣的canonical family，此外也會將一些Willmore猜想相關的性質附上並且補上他們的證明。

關鍵字

威爾莫猜想 ； 最小最大定理 ； 極小曲面

並列摘要

In 2012, the thesis: Min-max theory and Willmore conjecture wrote by Fernando C.Marques and Andre Neves. Which successful proof the well-known Willmore conjecture.
They used the method of geometric measure theory and application of Min-max theory gives a nice proof, which is how to create a special ability to use Min-max theory of canonical family, this is the whole thesis is the most important part; we will explain how to create such a canonical family in this article; on the other hand, we will addition to some property of the Willmore conjecture and their proof.

並列關鍵字

Willmore conjecture ； min-max theory ； minimal surface

參考文獻

F. Coda Marques and A. Neves, Min-max theory and the Willmore conjecture. To appear in Annals of Mathematics.

A. Ros, The Willmore conjecture in the real projective space, Math. Res. Lett. 6(1999),487-493.

P. Topping, Towards the Willmore conjecture, Calc. Var. Partial Differential Equations 11 (2000), 361-393.

F. Urbano, Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc. 108 (1990), 989-992.

L.Simon , Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra, (1983).