Minimal Surface Theory
Minimal Surfaces are area minimizing surfaces and therefore have vanishing mean curvature. Topics we will cover:
1. Computation of first variation of area.
2. Monotonicity formula.
3. Bernstein theorem, that complete graphs that are area minimizing in R^3 are flat planes. Global Isothermal coordinates.
4. Weierstrass parametrization via meromorphic functions of classical minimal surfaces.
5. Construction of minimal surfaces, Catenoids, Enneper surface, Scherck surface(which is periodic) using the Weierstrass parametrization.
6. Using Runge's theorem in complex analysis and the work of Jorge and Xavier and Nadirashvillli to construct a bounded, complete, minimal surface---a conjecture of ST Yau.
7. Omission of points by the Gauss map of complete, minimal surfaces. The theorems of Osserman, Fujimoto and Xavier.
8. Second variation formula for the Area functional.
9. Stability theorems for minimal surfaces, the theorems of D. Fischer-Colbrie and R. Schoen and do Carmo-Peng.
10. Setting up the Plateau problem and solving it. Morrey's \epsilon conformality theorem.
11. Theorems on total curvature of Cohn-Vossen and Huber, application to Minimal surfaces and relation to winding number of the non-compact ends. Costa's surface.
12. Survey of the regularity theory of minimal surfaces. Theorems of Osserman and others.
I have notes for this course. No textbook.
501, 502, 503.