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研究生:邱詩凱
研究生(外文):Shih-Kai Chiu
論文名稱:特殊拉格朗日球面存在性問題之探討
論文名稱(外文):On the Existence Problem of Special Lagrangian Spheres
指導教授:王金龍王金龍引用關係
口試委員:張樹城蔡忠潤
口試日期:2014-07-14
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2014
畢業學年度:102
語文別:英文
論文頁數:36
中文關鍵詞:特殊拉格朗日子流形瑞奇平坦度量
外文關鍵詞:special Lagrangian submanifoldsRicci-flat metrics
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在 Seidel 的博士論文 [Sei97] 中,他與他的指導教授 Donaldson 證明,若一緊緻凱勒流形 (compact Kahler manifold) 擁有一個尋常退化 (ordinary degeneration),則此凱勒流形內存在拉格朗日球面 (Lagrangian sphere)。這個結果引發以下的延伸問題:如果此凱勒流形為一卡拉比 -丘流形 (Calabi-Yau manifold),我們是否能夠在其中找出一個特殊拉格朗日球面 (special Lagrangian sphere)?透過文獻回顧,我們將探討特殊拉格朗日子流形 (special Lagrangian submanifolds) 的基本知識,以及球面的切叢 (the cotangent bundle of sphere) 上的瑞奇平坦度量 (Ricci-flat metrics)。在論文的最後,我們透過均曲率流 (mean curvature flow) 來探討一維的情形。

In his PhD thesis[Sei97], Paul Seidel and his advisor Simon K. Donaldson gave two proofs showing that a vanishing cycle in a Kahler manifold admitting an ordinary degener- ation can be chosen to be Lagrangian. This gives rise to the question whether the vanishing cycle is special Lagrangian if the manifold is Calabi-Yau. We investigate this problem by reviewing the geometric aspect of special Lagrangian manifolds and the Ricci-flat met- rics on the noncompact local model, namely the cotangent bundle of sphere. Finally, we approach this problem in dimension one through mean curvature flow.

Contents
口試委員會審定書 i
謝辭 ii
中文摘要 iii
Abstract iv
1 Introduction 1
2 Special Lagrangian Geometry 2
2.1 Definitions and Basic Results........................ 2
2.2 McLean’s Theorem............................. 3
2.3 Geometric Structures on the Local Moduli Spaces . . . . . . . . . . . . . 9
3 Ricci-flat metrics on T^&;#8727; S^n 15
3.1 Existence of the Metric........................... 15
3.2 Completeness of the Stenzel Metric .................... 19
3.3 Special Lagrangian Structures ....................... 21
4 Existence of Lagrangian Spheres 23
4.1 Seidel’s Proof................................ 24
4.2 Donaldson’s Proof ............................. 27
5 Discussion on the Main Problem 29
5.1 Formulation of the Main Problem ..................... 29
5.2 Results in n=1............................... 30

[EH89] Klaus Ecker and Gerhard Huisken. Mean curvature evolution of entire graphs. Annals of Mathematics, pages 453–471, 1989.
[Hit97] Nigel Hitchin. The moduli space of special lagrangian submanifolds. arXiv preprint dg-ga/9711002, 1997.
[HL82] Reese Harvey and H Blaine Lawson. Calibrated geometries. Acta Mathematica, 148(1):47–157, 1982.
[Joy03] Dominic Joyce. Riemannian holonomy groups and calibrated geometry. Springer, 2003.
[Mar02] Stephen P Marshall. Deformations of special Lagrangian submanifolds. PhD thesis, University of Oxford, 2002.
[McL96] Robert C McLean. Deformations of calibrated submanifolds. In Commun. Analy. Geom. Citeseer, 1996.
[PW91] Giorgio Patrizio and Pit-Mann Wong. Stein manifolds with compact symmetric center. Mathematische Annalen, 289(1):355–382, 1991.
[Sei97] Paul Seidel. Floer homology and the symplectic isotopy problem. PhD thesis, University of Oxford, 1997.
[Smo96] Knut Smoczyk. A canonical way to deform a lagrangian submanifold. arXiv preprint dg-ga/9605005, 1996.
[Ste93] Matthew B Stenzel. Ricci-flat metrics on the complexification of a compact rank one symmetric space. Manuscripta Mathematica, 80(1):151–163, 1993.

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