Preface
1. Introduction
2. Basic results in classical minimal surface theory
2.1 Eight equivalent definitions of minimality
2.2 Weierstrass representation
2.3 Minimal surfaces of finite total curvature
2.4 Periodic minimal surfaces
2.5 Some interesting examples of complete minimal surfaces
2.6 Monotonicity formula and classical maximum principles
2.7 Ends of properly embedded minimal surfaces
2.8 Second variation of area, index of stability and Jacobi functions
2.9 Barrier constructions
3. Minimal surfaces with finite topology and more than one end
3.1 Classification results for embedded minimal surfaces of finite total curvature
3.2 Constructing embedded minimal surfaces of finite total curvature
4. Limits of embedded minimal surfaces without local area or curvature bounds
4.1 Colding-Minicozzi theory
4.1.1 Main strategy of the proof of Theorem 4.1.3.
4.1.2 More details on the main strategy
4.2 Minimal laminations with isolated singularities
5. The structure of minimal laminations of R3
6. The Ordering Theorem for the space of ends
7. Conformal structure of minimal surfaces
7.1 Recurrence and parabolicity for manifolds
7.2 Universal superharmonic functions
7.3 Quadratic area growth and middle ends
8. Uniqueness of the helicoid I: Proper case
9. Embedded minimal annular ends with infinite total curvature
9.1 Harmonic functions on annuli
9.2 Annular minimal ends of infinite total curvature
10. The embedded Calabi-Yau problem
10.1 Uniqueness of the helicoid II: Complete case
10.2 Regularity of the singular set of convergence of a limit minimal lamination
11. Local pictures, local removable singularities and dynamics
12. Embedded minimal surfaces of finite genus
12.1 The Hoffman-Meeks conjecture
12.2 Non-existence of one-limit-ended examples
12.3 Uniqueness of the Riemann minimal examples
12.4 Colding-Minicozzi theory (fixed genus)
13. Topological aspects of minimal surfaces
14. Partial results on the Liouville Conjecture
15. The Scherk Uniqueness Theorem
16. Calabi-Yau problems
17. Outstanding problems and conjectures
Bibliography
