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研究生:李國瑋
研究生(外文):Kuo-Wei Lee
論文名稱:極小子流形與均曲率流
論文名稱(外文):Minimal Submanifolds and Mean Curvature Flow
指導教授:李瑩英李瑩英引用關係
指導教授(外文):Yng-Ing Lee
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:2004
畢業學年度:92
語文別:英文
論文頁數:32
中文關鍵詞:極小子流形均曲率流
外文關鍵詞:mean curvature flowminimal submanifolds
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本論文之第一部份是比較極小子流形在餘維為 $1$ 與高餘維時性質的差異。對於 Dirichlet Problem 而言,餘維為 $1$ 時其解存在、唯一且解析。但在高餘維時通常不成立;另外探討 Bernstein Theorem,餘維為 $1$ 時,若大空間的維度過高 (大於 $8$),當曲面能寫成一個函數的圖形、且為完備極小子流形時,則必須對於函數的梯度有所假設才能推得此流形是平面;在高餘維時都必須要有額外的假設下定理才會成立。

第二部份是有關王慕道對於均曲率流的一些文章。在任意餘維下,當一個流形(若能寫成某個函數的圖形)沿著均曲率的方向變化,藉由特別的 parallel $n$-form 與一些假設下,希望此均曲率流也保持圖形的型式、保持距離與面積遞減之性質與長時間的存在性,並且希望流到一個極小子流形(收斂性)。
The study of minimal submanifolds has a long and rich history. When the minimal submanifold is of non-parametric form, many beautiful results have been proved in the case of codimension one. However, the situation in higher codimension is quite different and is much less studied.
The first part of this thesis summarizes and gives a comparison between the results in codimension one and higher codimension. Chapter 1, I introduce basic definitions and terminology as preliminaries. Results on the Dirichlet problem of minimal surface systems are discussed in chapter 2. In section 2.1, I summerize the results, and explain in more detail in section 2.2 and 2.3 about some counterexample in higher codimension. Another important aspect of minimal submanifolds, namely Bernstein
theorem, is studies in chapter 3. In the second part of this thesis, I study some resent papers of M. T. Wang.
With the aid of an additional form, he makes a big progress in mean curvature flow in higher codimension and proves many interesting results. The main theme of my study in master program is to understand his results and method.
In chapter 4, I introduce the the main idea of Wang''s work to fill in some missing arguments in his paper.
Table of Contents iv
Introduction v
1 Preliminaries 1
1.1 minimal surface equation (system) 1
2 Dirichlet Problem 3
2.1 Some facts about Dirichlet Problem 3
2.2 The non-uniqueness and non-stability of solutions of dimension two with higher codimension 4
2.3 The non-existence of solution when m > 1 7
3 Bernstein Theorem 10
4 Mean curvature flow 12
4.1 Mean curvature flow equations 12
4.2 Evolution equation for parallel n-form 13
4.3 Some results on mean curvature flow 17
Bibliography 30
1. Allard, W. K., On the first variation of a varifold., Ann of Math., 95 (1972), 417-491.
2. Almgren, F.J., it Some interior regularity theorems for minimal surfaces and an extension of Bernstein''s theorem., Ann of Math. (2), 84, 277-292 (1966)
3. Bernstein, S.N., Uber ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus., Math. Z. 26, 551-558 (1927)
4. De Giorgi, E., Una estensione del teorema di Bernstein., Ann. Scuola Norm. Sup. Pisa 19, 79-85 (1965)
5. K. Ecker and G. Huisken, A Bernstein result for minimal graphs of constrolled growth., J. Differential Geom. 31 (1990), no. 2, 397-400.
6. Finn, R., Remarks relevant to minimal surfaces and surfaces of prescribed mean curvature.} J. Analyse Math., 14 (1965),139-160
7. S. Hildebrandt, J. Jost and K.-O. Widman, Harmonic mapping and minimal submanifolds., Invent. Math. 62 (1980/81), no. 2, 269-298.
8. G. Huisken, asymptotic behavior for singularities of the mean curvature flow., J. Differential Geom. 31 (1990), no. 1, 285-299
9. Jenkins, H. & Serrin, J. The Dirichlet problem for the minimal surface equation in higher dimensions., J. Reine Angew. Math., 229 (1968), 170-187
10. J. Jost and Y. L. Xin, Bernstein type theorems for higher codimension., Calc. Var. Partial Differential Equations 9 (1999), no 4, 277-296.
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