A Course in Minimal Surfaces

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
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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
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