　　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zui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zui大值原理。

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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

前言/序言

　　The motivation for these lecture notes on minimal surfaces is to have a treatment that begins with almost no prerequisites and ends up with current research topics. We touch upon some of the applications to other fields including low dimensional topology， general relativity， and materials science.
　　Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces， estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work， and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces.
　　The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle. Of the various ways of approaching minimal surfaces (from complex analysis， PDE， or geometric measure theory)， we have chosen to focus on the PDE aspects of the theory.
　　In Chapter 1， we will first derive the minimal surface equation as the Euler-Lagrange equation for the area functional on graphs. Subsequently， we derive the parametric form of the minimal surface equation (the first variation formula). The focus of the first chapter is on the basic properties of minimal surfaces， including the monotonicity formula for area and the Bernstein theorem. We also mention some examples. In the next to last section of Chapter 1， we derive the second variation formula， the stability inequality， and define the Morse index of a minimal surface. In the last section， we introduce multi-valued minimal graphs which will play a major role later when we discuss results from [CM3]-[CM7l. We will also give a local example， from [CM18l， of spiraling minimal surfaces (like the helicoid) that can be decomposed into multi-valued graphs but where the rate of spiraling is far from constant.
　　Chapter 2 deals with generalizations of the Bernstein theorem. We begin the chapter by deriving Simons' inequality for the Laplacian of the norm squared of the second fundamental form of a minimal hypersurface ∑ in Rn.In the later sections， we discuss various applications of this inequality. The first application is a theorem of Choi and Schoen giving curvature estimates for minimal surfaces with small total curvature. Using this estimate， we give a short proof of Heinz's curvature estimate for minimal graphs. Next， we discuss a priori estimates for stable minimal surfaces in three-manifolds， including estimates on area and total curvature of Colding and Minicozzi and the curvature estimate of Schoen. After that， we follow Schoen， Simon and Yau and combine Simons' inequality with the stability inequality to show higher Lp bounds for the square of the norm of the second fundamental form for stable minimal hypersurfaces. The higher Lp bounds are then used together with Simons' inequality to show curvature estimates for stable minimal hypersurfaces and to give a generalization due to De Giorgi， Almgren， and Simons of the Bernstein theorem proven in Chapter 1. We introduce a notion of "almost stabilility" that plays a crucial role in understanding embedded surfaces. Next， we return to multi-valued minimal graphs and prove an important result from [CM3] which states that the separation grows sublinearly if the multi-valued graph has enough sheets. We close the chapter with a discussion of minimal cones in Euclidean space and the relationship to the Bernstein theorem.
　　We start Chapter 3 by introducing stationary varifolds as a generalization of classical minimal surfaces. We next prove the Sobolev inequality of Michael and Simon. After that， we prove a generalization， due to Colding and Minicozzi， of the Bernstein theorem for minimal surfaces discussed in the preceding chapter. Namely， following [CM6]， we will show in Chapter 3 that， in fact， a bound on the density gives an upper bound for the smallest affine subspace that the minimal surface lies in. We will deduce this theorem from the properties of the coordinate functions (in fact， more generally， properties of harmonic functions) on k-rectifiable stationary varifolds of arbitrary codimension in Euclidean space. Finally， in the last section， we introduce another notion of weak convergence (called bubble convergence) that was developed to explain the bubbling phenomenon that occurs in conformally invariant problems， including two-dimensional harmonic maps and J-holomorphic curves. We will show that bubble convergence implies varifold convergence.
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