250124 VO Topics in Geometric Analysis (2015S)

Syllabus:

Soap films have every intention to span their boundary with the least amount of area possible. The development of their mathematical theory and its generalization is a cornerstone of twentieth century mathematics. My goal in this class is to give a reasonably self-contained introduction to this theory of "minimal surfaces" and to illustrate its applications in several areas of mathematics possibly including a discussion of pseudoholomorphic curves and the theory of horizons in general relativity.

To get us started, I will review aspects of the theory of submanifolds of Euclidean space and Riemannian geometry. We then turn to the classical mathematical theory of soap films spanning a given boundary as developed by J. Douglas (who received one of the first two Fields medals for this contribution) and T. Rado. This discussion will contain a proof of the uniformization theorem in complex analysis as a special case. We then turn to properties of general minimal hypersurfaces including the crucial monotonicity formula and a discussion of minimal graphs. Our next goal is a a slick derivation of the curvature estimates for stable minimal hypersurfaces of E. Heinz, R. Schoen, L. Simon, and S.-T. Yau. This will be a crucial ingredient for an unusual proof of existence of area minimizing hypersurfaces spanning a given boundary in $\mathbb{R}^{n+1}$ where $2 \leq n \leq 4$. We will completely dodge geometric measure theory in this, but clearly recognize its advent as a necessary and logical development. The class will conclude with selected topics in minimal surface theory that suit the gusto of the audience.

Prerequisites:

It will be very useful to have familiarity with the basics of elliptic partial differential equations and differential geometry. The existence of solutions for the Dirichlet problem for the minimal surface equation will be used as a black box. You should be prepared to work very hard to take advantage of this class.