# 臺灣博碩士論文加值系統

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 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。
 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.
 誌謝………………………………………………………………………………… i中文摘要………………………………………………………………………… ii英文摘要………………………………………………………………………… iii第一章 Introduction…………………………………………………………… 1第二章 Some property of the Willmore energy……………………………… 3第三章 Min-max theory……………………………………………………… 7第四章 Construct the canonical family and Min-max theory………………… 9第五章 Proof of the main theorem…………………………………………… 22參考文獻………………………………………………………………………… 28
 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).T. J. Willmore, Note on embedded surfaces, An. Sti. Univ. "Al. I. Cuza" Iasi Sect. I a Mat. (N.S.) 11B (1965) 493-496.T. J. Willmore, Riemannian Geometry, Oxford, England : Clarendon Press ; New York : Oxford University Press, 1993M. Bauer and E. Kuwert, Existence of minimaizing Willmore surfaces of prescribed genus, Int. Math. Res. Not. (2003), 553-576.B. Lawson, Complete minimal surfaces in S3, Ann. of Math. (2) 92 (1970).E.Kuwert, Y.Li, and R. Schatzle, The large genus limit of the infimum of the Willmore energy, Amer. J. Math. 132 (2010), 37-51.T. J. Willmore, Mean curvature of Riemannian immersions, J. London Math. Soc.(2) 3 1971 307-310.K.Shiohama and R. Takagi, A characterization of a standard torus in E3, J. Differential Geometry 4 1970 477-485.B-Y. Chen, On the total curvature of immersed manifolds. VI. Submanifolds of finite type and their applications, Bull. Inst. Math. Acad. Sinica 11 (1983), 309-328.J. Langer and D. Singer, Curves in the hyperbolic plane and mean curvature of tori in 3-space, Bull. London Math. Soc. 16 (1984), 531-534.P.Li and S-T. Yau , A new conformal invariant and its applications to the Wilmore conjecture and the first eigenvalue of compact surface , Invent. Math. 69(1982)
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