The Theory of Minimal Submanifolds: The equations of minimal submanifolds and examples. The first and second variation formulas of volume, and stability. Bernsteins’ theorem, the Weierstrass representation and the strong maximum principle. Multi-valued minimal graphs.
Curvature Estimates: Simons, inequality. Small energy curvature estimates for minimal surfaces. L^p curvature bounds for stable hypersurfaces. Almost stability. Minimal cones.
Existence of Minimal Surfaces: The Plateau problem and its solution. Harmonic maps. Existence of minimal spheres in a homotopy class.
Minimal Surfaces in Three-Manifolds: Hersch’s and Yang-Yau’s Theorem. The Reilly Formula. Choi and Wang’s estimate. Compactness Theorems with a priori bounds. The positive mass theorem.
Colding-Minicozzi Theory: The structure of embedded minimal disks. Compactness Theorems without a priori bounds.
COLDING, T, H., MINICOZZI, WILLIAM P. – A course in minimal surfaces. Providence, R.I.: American Mathematical Society, c2011. xii, 313 p. (Graduate studies in mathematics ; v. 121).