极小曲面教程
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作者(美)托拜厄斯·柯丁,(美)威廉·米尼科帝二世 著
出版社高等教育出版社
ISBN9787040469110
出版时间2017-01
装帧精装
开本16开
定价135元
货号1201480943
上书时间2024-12-05
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目录
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.SimonsInequality
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.Birkhoffs 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.Herschs and Yang and Yaus Theorems
3.The Reilly Formula
4.Choi and Wangs 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
内容摘要
极小曲面可追溯到欧拉和拉格朗日以及变分法发轫的年代，它的很多技术在几何和偏微分方程中发挥着关键作用，例子包括：源自极小曲面正则性理论的单调性和切锥分析，基于Bernstein的经典工作最大值原理的非线性方程估值，还有勒贝格的积分定义——这是他在有关极小曲面的Plateau问题的论文中发展出来的。
《极小曲面教程（英文版）》从极小曲面的经典理论开始，以当今的研究专题结束。在处理极小曲面的各种方法（复分析、偏微分方程或者几何测度论）中，作者选择了将注意力放在这个理论的偏微分方程方面。《极小曲面教程（英文版）》也包含极小曲面在其他领域的应用，包括低维拓扑、广义相对论以及材料科学。
《极小曲面教程（英文版）》的预备知识仅要求了解黎曼几何的基本知识并熟悉最大值原理。
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