Course Descriptions

16:640:519 - Selected Topics in Differential Equations

Sagun Chanillo


Topics in Minimal Surfaces


No text book, though R. Osserman's book, in Dover Press is good for some topics


Real analysis, undergraduate differential geometry


This course will be an introduction to minimal surfaces, which are naively area minimizing hypersurfaces. This is usually called soap film geometry. I shall cover, the (1) first variational formula for area for a surface, (2)Weierstrass representation and construction of minimal surfaces, (3) monotonicity formula, (4) Bernstein's theorem that a complete minimal graph has to be a flat plane, (5) The Gauss map, if the Gauss map of a complete minimal surface omits 9 points on S^2, then it is a flat plane (6) Plateau problem (7) Second variational formula and stability of minimal surfaces theorem of R. Schoen and Fischer-Colbrie, Do Carmo and Peng that a complete, stable minimal surface is a flat plane (8) Curvature estimates of Heinz and its generalization by Schoen-Simon-Yau (9) Morse theory of constant mean curvature surfaces the paper of Chanillo-Malchiodi, Comm. in Analysis and Geometry 13(1) (2005) 187-251.

Schedule of Sections: