極小曲面的極小極大構造法

Min-max construction of minimal surface

黃垣熊(Yuan Shyong Ooi)

指導教授 ： 李瑩英

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.02256

摘要

本論文將討論如何通過極小極大方法來構造極小曲面。我們主要討論在這方法之下，極小曲面的存在性問題。
這方法有很多不同的版本，我們主要的參考文獻來自
Colding 和 De Lellis 的 The min-max construction of minimal
surfaces[CD]。我們也將會在第一章節提到其他造極小曲面的極小極大方法。

關鍵字

極小極大構造法 ； 極小曲面

並列摘要

In this thesis, we shall survey the construction of minimal surface in closed three-manifold via min-max construction.
Our focus will be on the existence of the min-max stationary varifold via this construction. There are many different type min-max construction. Our main reference is The min-max construction of minimal surfaces [CD] by Colding and De Lellis in which they apply min-max method in the isotopy class of generalized family of surfaces. We shall also mention some other min-max method in the introduction part.

並列關鍵字

min-max construction ； minimal surfaces

參考文獻

[All] W.K. Allard, On the first variation of a varifold, Ann.of Math.(2) 95 (1972) no. 3, 417--491.

[CP] C. De Lellis and F. Pellandini, Genus bounds for minimal surfaces arising from min-max constructions, J.Reine Angew. Math. 644(2010), 47--99.

[J]J. Jost, Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere, J. Differential Geom. 30(1989), no. 2, 555--577.

[MSY]W. Meeks III, L. Simon, and S.T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive ricci curvature Ann.of Math.(2) 116 (1982) no. 3, 621--659.

[MN] F. C. Marques and A. Neves, Rigidity of min-max minimal spheres in three-manifolds, Duke Mathematical Journal 161 (2012), no. 14,2725-2752.