Contents
Preface ix
Chapter 1. The Beginning of the Theory 1
§1. The Minimal Surface Equation and Minimal Submanifolds 1
§2. Examples of Minimal Surfaces in
R3
8
§3. Consequences of the First Variation Formula 18
§4. The Gauss Map 28
§5. The Theorem of Bernstein 29
§6. The Weierstrass Representation 33
§7. The Strong Maximum Principle 37
§8. Second Variation Formula, Morse Index, and Stability 38
§9. Multi-valued Graphs 50
§10. Local Examples of Multi-valued Graphs 51
Appendix: The Harnack Inequality 60
Appendix: The Bochner formula 60
Chapter 2. Curvature Estimates and Consequences 65
§1. Simons’ Inequality 66
§2. Small Energy Curvature Estimates for Minimal Surfaces 72
§3. Curvature and Area 76
§4.
Lp
Bounds of
|A|2
for Stable Hypersurfaces 85
§5. Bernstein Theorems and Curvature Estimates 87
§6. The General Minimal Graph Equation 88
§7. Almost Stability 92
v
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