卡拉比 -丘流形中的拉格朗日和特殊拉格朗日子流形

Lagrangian and Special Lagrangian Submanifolds in a Calabi-Yau Manifold

蘇瑋栢(Wei-Bo Su)

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.2014.00973

摘要

第一部分把拉格朗日子流形的幾何和一些已知的結果做一個大略的 簡介,文中特別討論了 Hitchin 對於特殊拉格朗日子流形的模空間結構 的研究結果。
第二部分開始利用均曲率流來研究特殊拉格朗日子流形在卡拉 比 -丘空間中的存在性。除了簡介一些已知結果外,我們會在文中仔 細討論由 Joyce-Lee-Tsui 所給出的自相似解。最後我們會簡介一個由 Thomas-Yau 給出的一個關於特殊拉格朗日子流形存在性的猜想。

關鍵字

均曲率流 ； 拉格朗日子流形 ； 特殊拉格朗日子流形

並列摘要

In the first part of this thesis, we survey and summarize some known re- sults in Lagrangian and special Lagrangian geometry. In particular we fo- cus on the structure of the moduli space of special Lagrangian subamanifolds studied by Hitchin.
In the second part, we focus on the theory of Lagrangian mean curvature flow, especially the Lagrangian self-similar solutions constructed by Joyce- Lee-Tsui. At the end we describe an unsolved conjecture given by Thomas and Yau.

並列關鍵字

mean curvature flow ； Lagrangian submanifold ； special Lagrangian submanifold

參考文獻

[CL04] Jingyi Chen and Jiayu Li. Singularity of mean curvature flow of Lagrangian submanifolds. Invent. Math., 156(1):25–51, 2004.

[Gro99] Mark Gross. Special Lagrangian fibrations. II. Geometry. A survey of tech- niques in the study of special Lagrangian fibrations. In Surveys in differential geometry: differential geometry inspired by string theory, volume 5 of Surv. Differ. Geom., pages 341–403. Int. Press, Boston, MA, 1999.

[JLT10] Dominic Joyce, Yng-Ing Lee, and Mao-Pei Tsui. Self-similar solutions and translating solitons for Lagrangian mean curvature flow. J. Differential Geom., 84(1):127–161, 2010.

[Joy03a] Dominic Joyce. Singularities of special Lagrangian fibrations and the SYZ conjecture. Comm. Anal. Geom., 11(5):859–907, 2003.

[Joy03b] Dominic Joyce. Special Lagrangian submanifolds with isolated conical sin- gularities. V. Survey and applications. J. Differential Geom., 63(2):279–347, 2003.