Instructor: J. Douglas Moore
Prerequisites: Math 221ABC (Foundations of topology) or Math 240ABC (Differential and Riemannian
Geometry) or consent of instructor.
During the last half of the twentieth century, differential geometry was heavily influenced
by new ideas coming from nonlinear partial differential equations. First, there was the theory
of minimal surfaces in Riemannian manifolds, which in the hands of Meeks, Schoen and Yau led to
advances in three-dimensional topology and a resolution of the positive mass conjecture from general
relativity in 1979. Next the standard model for elementary particles in physics suggested that geometers study
the Yang-Mills equation. This led to Donaldson's theory which showed that some topological
four-manifolds (such as four-dimensional Euclidean space) have many different smooth structures.
Donaldson's theory was later simplified immensely by means of the Seiberg-Witten equations in the mid-90's.
Finally, there was a revolution in symplectic geometry stimulated by Gromov's theory of pseudoholomorphic
curves.
What all of these developments have in common is this---a global analysis approach to partial
differential equations that arise from geometry---the minimal surface equation, the Yang-Mills
equation, the Seiberg-Witten equations or the equations for pseudoholomorphic curves. In technical
terms, these techniques work because all of these equations lie on the "border of Sobolev range."
In some sense, the Morse theory of geodesics and minimal surfaces provides the simplest context for
understanding the foundational techniques of global analysis that were developed by Smale and
others to study such nonlinear PDE's.
This course will be devoted to the Morse theory of geodesics and two-dimensional minimal surfaces,
as foreshadowed by Smale's celebrated 1964 paper, "Morse theory and a nonlinear generalization of the
Dirichlet problem." But much of the course will be devoted to the basic tools of global analysis that are
needed for all of the applications mentioned above: these tools are infinite-dimensional manifolds,
transversality theory, elementary topology of spaces of maps including de Rham cohomology, and techniques
for applying these ideas to nonlinear PDE's.
The ultimate goal of the course will be to introduce techniques that are being developed to apply a partial
Morse theory of minimal surfaces to problems arising from Riemannian geometry.
An application is the sphere theorem of Micallef and Moore (1988): A compact simply connected manifold with
positive isotropic curvature is homeomorphic to a sphere. This theorem extended the earlier sphere theorems
of Berger, Klingenberg and Toponogov, and showed that a new type of curvature, isotropic curvature, is relevant
to the study of minimal surfaces.
Towards the end of the course, we will describe recent progress on establishing a partial Morse theory for
closed minimal surfaces in compact Riemannian manifolds. This Morse theory should enable one to use techniques
from topology (such as de Rham cohomology) to provide information on the simplest nonlinear partial differential equations which
arise in geometry.
We will begin with calculus on manifolds modeled on Banach or Hilbert spaces. Except for the fact that we
need to use some basic theorems from analysis (the Hahn-Banach theorem, the closed graph theorem and the
Baire category theorem), calculus on infinite-dimensional manifolds is mostly parallel to the
finite-dimensional theory.
The main examples of infinite-dimensional manifolds that we will use are spaces of maps, such as the manifold
Map(M,N) of maps f: M -> N, where M and N are finite-dimensional manifolds. We will develop the de Rham
cohomology theory for Map(M,N) and show how to calculate it in some instances using a theory of "minimal
models" due to Sullivan.
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. A celebrated conjecture of Klingenberg states that a compact
manifold with finite fundamental group has infinitely many geometrically distinct smooth closed geodesics. We
will give a generic version of a theorem of Gromoll and Meyer which proves Klingenberg's conjecture for many
manifolds, and explain how Sullivan's method of minimal models applies to solve the associated topological problem.
This sets the stage for a study of the corresponding problem for minimal surfaces: Find topological
conditions on a compact manifold M with finite fundamental group that ensure that the manifold has infinitely
many geometrically distinct closed minimal surfaces of a given genus for generic choice of Riemannian
metric on M. We will prove the theorem of Sacks and Uhlenbeck that any compact Riemannian manifold with finite
fundamental group contains at least one minimal sphere. The course will then conclude with recently developed
techniques for proving existence of minimax minimal tori, and a proof of the Micallef-Moore sphere theorem, as
well as a description of some open problems.
1. J. Milnor, Morse Theory, Princeton Univ. Press, Princeton NJ, 1963.
2. R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982), 331-358.
3. S. Lang, Differential and Riemannian Manifolds, Springer, New York, 1995.
4. S. Smale, Morse theory and a nonlinear generalization of the Dirichlet problem, Annals of Math, 80 (1964), 382-396.
5. R. Bott and L. Tu, Differential forms in algebraic topology, Springer, New York, 1982.
Items 1, 3 and 5 will be available from the campus book store and will also be put on reserve at the campus library. Items Reference 4 can be accessed via Math Sci Net: http://www.ams.org/mathscinet/
Every student who is interested in writing a dissertation in topology or differential geometry should read the first
six sections of Milnor's classic book on Morse Theory (38 pages).
Some students may also benefit from Bott's alternative treatment, which is provided in pages 331-350 of his 1982 survey
article in the Bulletin of the American Mathematical Society.
Lang's book on Differential and Riemannian manifolds is justly famous; an earlier version was the first treatment in
book form of manifolds modeled on infinite-dimensional Banach spaces. We will follow his treatment (particularly
Chapters 1 and 2) to some extent in the early part of the course.
Smale's article on Morse theory shows how to extend the theory presented in Part I of Milnor's Morse theory to infinite-
dimensional manifolds. We will study pages 382-387 of this article in the second quarter of the course.
Students not familiar with de Rham theory (the simplest approach to cohomology theory) may benefit by reading the
Introduction and sections 1-5 of the book by Bott and Tu (pages 1-42). Section 19 of this book also gives a very brief
introduction to Sullivan's theory of minimal models. We will present a somewhat more detailed discussion of Sullivan's
theory in the second quarter of the course.
During the course, we will develop a set of lecture notes which will be made available in PDF format at the link
Lecture notes on Morse theory, which can be accessed by
terminals on the UCSB campus. The lecture notes at this link will be somewhat different from the lectures on some
topics, and will change as they are revised during the course.
If you would like me to write a strong letter of recommendation for you after the course is over, give a lecture during the course on some topic
related to Morse theory or minimal surfaces.