Course Outline for Math 241C (Spring 2014)

Minimal Surfaces in Riemannian Manifolds

 

 

Instructor: J. Douglas Moore

 

 

This course will develop calculus on manifolds which are modeled on Banach or Hilbert spaces. Except for the fact that we need to use some basic theorems from analysis, calculus on infinite-dimensional manifolds is mostly parallel to the finite-dimensional theory.  And infinite-dimensional manifolds are useful for studying many of the nonlinear differential equations that arise in geometry, such as the Yang-Mills equation, the Seiberg-Witten equations and the equations for minimal surfaces in a Riemannian manifold, which is the main topic of the course.

For the theory of minimal surfaces, the examples of infinite-dimensional manifolds that are important are spaces of maps, such as the manifold Map(M,N) of maps  f: M -> N,  where  M  and  N  are finite-dimensional manifolds.

Smooth closed geodesics on a Riemannian manifold M can be regarded as critical points for the action function  J : Map(C,M) -> R,  where  C  is the unit circle.  We will use Morse theory of  J  to give a generic version of a theorem of Gromoll and Meyer which shows that most compact manifold must have infinitely many geometrically distinct smooth closed geodesics for generic metrics.

This sets the stage for studying corresponding problems for minimal surfaces.  We will develop Morse theory for the alpha-energy on  Map(S,M),  where  S  is a surface and use it to prove the theorem of Sacks and Uhlenbeck that any compact Riemannian manifold with finite fundamental group contains at least one minimal two-sphere, as well as theorems of Schoen and Yau on existence of minimal surfaces of higher genus.  We will also explore the relationship between isotropic curvature and minimal surfaces.

Lecture notes:  During the course, we will develop a set of lecture notes which will be made available in PDF format on gauchospace.  The lecture notes will be revised as the course progresses.