第一海森堡群流形的伯恩斯坦定理

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姓名	楊健男(Chien-nan Yang) 查詢紙本館藏	畢業系所	數學系
論文名稱	第一海森堡群流形的伯恩斯坦定理 (Survey of Bernstein Type Theorems in the First Heisenberg Group)
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摘要(中)	為了引起大家的興趣，我們首先從現實生活中的概念來介紹海森堡群的定義。衛星導航是最近行車配備科技上最新的發展，也慢慢成為不可或缺的工具。假設我們從高空的衛星觀察高速公路上行駛的跑車，你會發現對於車子的位置和速度，兩者總會有其一並無法同時被描述地很清楚。在衛星上，我們只看到車子的位置，因為車子的移動量實在太小了。但對於路上的警察，他所能看到的是跑車從他面前呼嘯而過，至於他和跑車在地球上的位置，如果沒有地圖，他根本也不清楚。這個現象就是著名的海森堡測不準原理。當我們考慮原子尺度大小的事物就必須注意到這個現象。地球的尺度之於足球就好比是足球之於原子的比例一樣。事實上，由於電子的移動速率過快，真正的理論應該還須考慮進相對論的假設，但為了不增加複雜度，我們只暫時考慮海森堡群作為背景空間流形的模型。這篇概論性論文分成兩個主要的部分，第一個部分是關於伯恩斯坦(Bernstein)定理的概念及證明。最早是源自於肥皂泡泡膜的表面張力問題，表面張力是一種物理效應，它使得液體的表面總是試圖獲得最小的、光滑的面積，就好像它是一層彈性的薄膜一樣。其原因是液體的表面總是試圖達到能量最低的狀態。接下來第二個部分則是結合上述的海森堡模型，考慮海森堡群裡極小表面積曲面的問題，但由於在古典的情形中，我們刻劃極小表面積曲面的方式是藉由平均曲率為零的偏微分方程式，在海森堡群的情下並不適用，所以必須改用別的方式(variation method)來處理，最終我們整理出許多不同類型的伯恩斯坦形式的定理。但都只是在最簡單海森堡群上的結果，即變數只有位置、速度、時間三個分量的情形，在物理上有廣義位置向量、廣義速度向量、時間所形成的高維度相空間，同理，也有高維度的海森堡群，但伯恩斯坦問題在某些高維度的情形依然是還未解決的，因此這篇論文短期的目標是在提供有興趣的學者在最低維度情形下的導引，並希望未來可以理解在物理學上實際的對應及應用。
摘要(英)	To motivate the definition of the Heisenberg group H1, we briefly describe its origin from physics. Let’s assume that you are watching high-street traffic from an airplane. You will notice the position of the cars but you cannot say too much about their speed. They look like they are not moving at all. A policeman on the road will see the picture completely different. For him, the speed of the cars will make more sense than their position. The latter is changing too fast to be noticed accurately. The Heisenberg uncertainty principle can be found in the study of quantum particles at the large scale structure. This survey can be divided into two pars. The first part is about the notion and the proof of the classical Bernstein problem. It arises from the surface tension problem of soap film. In Mathematics, such an problem will be characterized by minimal surface equation. The second part is to combine the Heisenberg group model and the minimal surface equation. But the classical method is not suitable for use in the Heisenberg group case. Instead, we adopt the variational method to characterize the horizontal minimal surfaces. The investigate of the Bernstein problem on the Heisenberg group is an important tool to explore horizontal minimal surfaces. The goal of the survey of these recent results is to provide readers who may want to investigate this problem which is still open for Hn, n=2, 3, 4.
關鍵字(中)	★ 海森堡群流形 ★ 伯恩斯坦定理	關鍵字(英)	★ Bernstein theorem ★ Heisenberg group
論文目次	1. Introduction………………………………………………… 1 2. Classical Bernstein Problem………………………………… 2 2-1 Historical Survey…………………………………………… 2 2-2 Main Theorem……………………………………………… 4 3. The Higher Dimensional Classical Bernstein Theorem…… 9 4. Geometric Mechanics of the Heisenberg Group…………… 14 4-1 Definition for the First Heisenberg Group………………… 15 4-2 The Horizontal Distribution………………………………… 16 4-3 The Horizontal Gauss Map………………………………… 16 4-4 Horizontal Levi-Civita Connection………………………… 17 4-5 Horizontal Second Fundamental Form……………………… 17 5. Some Interesting Candidates for Minimal Surfaces in H1…… 19 6. Another Characterization of Minimal Surfaces…………… 20 6-1 First and Second variation of the H-perimeter……………… 21 6-2 The stability of H-minimal surfaces………………………… 22 7. Several Bernstein Type Theorems…………………………… 23 7-1 Entire stable minimal graphs with empty singular set……… 25 7-2 Entire stable minimal graphs with nonempty singular set… 26 7-3 Entire stable area-stationary graphs with nonempty singular set…………………………………………………………… 28 7-4 Complete stable surfaces with empty singular set………… 29 7-5 Complete stable surfaces with nonempty singular set……… 33 References ……………………………………………………………… 34
參考文獻	E. Bombieri, E. De Giorgi, E. Giusti, Minimal cones and the Bernstein problem, Inv. Math 7 (1969) 243-268. V. Barone Adesi, F. Serra Cassano and D. Vittone, The Bernstein problem for intrinsic graphs in the Heisenberg group and calibrations, Calc. Var. (2007) 30:17-49. J.H. Cheng, J. F. Hwang, A. Malchiodi, P. Yang, Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group, Ann. Sc. Norm. Sup. Pisa, 1~(2005), 129-177. D. Danielli, N. Garofalo, D. M. Nhieu, Sub-Riemannian calculus on hypersurfaces in Carnot groups, Adv. in Math. 215(2007) 292-378. D. Danielli, N. Garofalo, D. M. Nhieu, S. D. Pauls, Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group $HH$},JDG. 81(2009) 251-295, April 2006. D. Danielli, N. Garofalo, D. M. Nhieu, S. D. Pauls, The Bernstein Problem for Embedded Surfaces in the Heisenberg Group H1, Indiana University Mathematics Journal Vol.59, No. 2(2010). U. Dierkes, S. Hildebrandt, A.J. Tromba, Global Analysis of Minimal Surfaces, Springer-Verlag Berlin Heidelberg 2010. U. Dierkes, S. Hildebrandt, A. Kuster, O. Wohlrab Minimal Surfaces I, Grundlehren der math. Wiss. 295 Springer-Verlag Berlin Heidelberg 1992. A.I. Fomenko, Variational Principles in Topology-Multidimensional Minimal Surface Theory, 1990. W.H. Fleming, On the oriented Plateau problem, Rend, Circ, Mat. Palermo 11(1969), 69-90. N. Garofalo, S. D. Pauls, The Bernstein problem in the Heisenberg group, preprint, May 2005. R. Osserman, A survey of minimal surfaces}, Stanford University. S. Pauls, Minimal surfaces in the Heisenberg group, Geom. Dedicata., 104~(2004), 201-231. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math.88 (1968), 62-105 . A. Hurtado, M. Ritore, C. Rosales The Classification of complete stable area-stationary surfaces in the Heisenberg group H1, Adv. in Math. 224(2010) 561-600. M. Ritore, C. Rosales, Area stationary surfaces in the Heisenberg group H1,Adv. in Math. 219(2008) 633-671.
指導教授	饒維明(D. M. Nhieu)	審核日期	2012-7-16
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博碩士論文 942201027 詳細資訊