極小曲面教程（英文版）

定　價(jià)：￥135.00

作　者：	[美] 托拜厄斯·柯?。═obias，Holck，Colding，），美]威廉·米尼科帝二世著
出版社：	高等教育出版社
叢編項(xiàng)：	美國數(shù)學(xué)會經(jīng)典影印系列
標(biāo)　簽：	數(shù)學(xué) 數(shù)學(xué)理論自然科學(xué)

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ISBN：	9787040469110	出版時(shí)間：	2017-01-01	包裝：	精裝
開本：	16開	頁數(shù)：	313	字?jǐn)?shù)：

內(nèi)容簡介

　　極小曲面可追溯到歐拉和拉格朗日以及變分法發(fā)軔的年代，它的很多技術(shù)在幾何和偏微分方程中發(fā)揮著關(guān)鍵作用，例子包括：源自極小曲面正則性理論的單調(diào)性和切錐分析，基于Bernstein的經(jīng)典工作zui大值原理的非線性方程估值，還有勒貝格的積分定義——這是他在有關(guān)極小曲面的Plateau問題的論文中發(fā)展出來的?！稑O小曲面教程（英文版）》從極小曲面的經(jīng)典理論開始，以當(dāng)今的研究專題結(jié)束。在處理極小曲面的各種方法（復(fù)分析、偏微分方程或者幾何測度論）中，作者選擇了將注意力放在這個(gè)理論的偏微分方程方面?！稑O小曲面教程（英文版）》也包含極小曲面在其他領(lǐng)域的應(yīng)用，包括低維拓?fù)?、廣義相對論以及材料科學(xué)?！稑O小曲面教程（英文版）》的預(yù)備知識僅要求了解黎曼幾何的基本知識并熟悉zui大值原理。

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圖書目錄

Preface
Chapter 1． The Beginning of the Theory
1．The Minimal Surface Equation and Minimal Submanifolds
2．Examples of Minimal Surfaces in R3
3．Consequences of the First Variation Formula
4．The Gauss Map
5．The Theorem of Bernstein
6．The Weierstrass Representation
7．The Strong Maximum Principle
8．Second Variation Formula， Morse Index， and Stability
9．Multi-valued Graphs
10．Local Examples of Multi-valued Graphs
Appendix： The Harnack Inequality
Appendix： The Bochner formula
Chapter 2． Curvature Estimates and Consequences
1．Simons'Inequality
2．Small Energy Curvature Estimates for Minimal Surfaces
3．Curvature and Area
4．Lp Bounds of |A|2 for Stable Hypersurfaces
5．Bernstein Theorems and Curvature Estimates
6．The General Minimal Graph Equation
7．Almost Stability
8．Sublinear Growth of the Separation
9．Minimal Cones
Chapter 3． Weak Convergence， Compactness and Applications
1．The Theory of Varifolds
2．The Sobolev Inequality
3．The Weak Bernstein-Type Theorem
4．General Constructions
5．Finite Dimensionality
6．Bubble Convergence Implies Varifold Convergence
Chapter 4． Existence Results
1．The Plateau Problem
2．The Dirichlet， Problem
3．The Solution to the Plateau Problem
4．Branch Points
5．Harmonic Maps
6．Existence of Minimal Spheres in a Homotopy Class
Chapter 5． Min-max Constructions
1．Sweepouts by Curves
2．Birkhoff's Curve Shortening Process
3．Existence of Closed Geodesics and the Width
4．Harmonic Replacement
5．Minimal Spheres and the Width
Chapter 6． Embedded Solutions of the Plateau problem
1．Unique Continuation
2．Local Description of Nodal and Critical Sets
3．Absence of True Branch Points
4．Absence of False Branch Points
5．Embedded Solutions of the Plateau Problem
Chapter 7． Minimal Surfaces in Three-Manifolds
1．The Minimal Surface Equation in a Three-Manifold
2．Hersch's and Yang and Yau's Theorems
3．The Reilly Formula
4．Choi and Wang's Lower Bound for λ1
5．Compactness Theorems with A Priori Bounds
6．The Positive Mass Theorem
7．Extinction of Ricci Flow
Chapter 8． The Structure of Embedded Minimal Surfaces
1．Disks that are Double-spiral Staircases
2．One-sided Curvature Estimate
3．Generalized Nitsche Conjecture
4．Calabi-Yau Conjectures for Embedded Surfaces
5．Embedded Minimal Surfaces with Finite Genus
Exercises
Bibliography
Index