極小子流形與均曲率流

Minimal Submanifolds and Mean Curvature Flow

李國瑋(Kuo-Wei Lee)

指導教授 ： 李瑩英

國立臺灣大學/理學院/數學研究所/碩士(2004年)

https://doi.org/10.6342/NTU.2004.00386

摘要

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)。

關鍵字

極小子流形 ； 均曲率流

並列摘要

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.

並列關鍵字

mean curvature flow ； minimal submanifolds

參考文獻

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.