指導教授：王金龍臺灣大學：數學研究所邱詩凱Chiu, Shih-KaiShih-KaiChiu2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264028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^∗ 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............................... 30449213 bytesapplication/pdf論文公開時間：2014/08/25論文使用權限：同意無償授權特殊拉格朗日子流形瑞奇平坦度量thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264028/1/ntu-103-R01221031-1.pdf