Abstract:

Morse theory might have developed in three stages. The first stage would have described the relationship between critical points on a finite dimensional manifold and the topology of the manifold. The second might have been "calculus of variations in the large" for ODE's including the Morse theory of geodesics. Palais and Smale formulated this in terms of infinite-dimensional manifolds, and Smale expressed the hope that this approach might lead to a similar theory for minimal surfaces, and related PDE's. Sacks and Uhlenbeck described how to perturb the theory of minimal surfaces and develop a Morse theory for the perturbed problem, initiating the third stage.

In this talk I will describe what happens when the perturbation is turned off. One gets a partial Morse theory in the limit. Applications to the existence of minimal tori in Riemannian manifolds will be presented.