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.