MATH 521: Topics in Real Analysis, Fall 2000

Nonpositively curved spaces, harmonic maps, and some applications to group actions

Description:

Harmonic maps into Riemannian manifolds of nonpositive sectional curvature enjoy various existence, uniqueness, and regularity properties. These were well-studied for two decades starting in the sixties. Nonpositive curvature also occurs naturally in many symmetric spaces under various group actions. With dimension greater than 2, there are often rigidity results (e.g. Mostow rigidity) wherby actions by isomorphic groups or the presence of diffeomorphic quotients leads to the existence of an isometry between the quotients. New proofs of both old and new rigidity results use harmonic maps to certain natural quotient metric spaces which are singular, but negatively-curved in some generalized sense. These proofs require some new analytic and geometric machinery, which is of interest beyond the rigidity questions.

Some of the topics we hope to cover include:

Some examples of rigidity of geometric structures and group actions
Properties of nonpositively-curved manifolds
Properties of nonpositively-curved metric spaces
Finite energy maps to nonpositively-curved spaces
Existence of harmonic maps
Conditions implying that harmonic maps are totally-geodesic
Rigidity results
The cone at infinity

Prerequisites for the course include some knowledge of Riemannian geometry.

Meets:

MWF 1PM in Herman Brown 22

Instructor:

Robert Hardt Office: Herman Brown 430; Office hours: 11-12 MWF (and others by appt.),
Email: hardt@rice.edu, Telephone: ext 3280
http://math.rice.edu/~hardt/521F00/

References:

J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, -ETH Lectures in Mathematics, Birkhauser Verlag Basel, 1997.

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Notes 25, Birkhauser Verlag,, Basel, 1995.

M.Gromov & R.Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. No. 76, (1992), 165--246.

N.Korevaar & R.Schoen, Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1 (1993), no. 3-4, 561--659.

N. Korevaar & R.Schoen, Global existence theorems for harmonic maps to non-locally compact spaces.Comm. Anal. Geom. 5 (1997), no. 2, 333--387.

N. Korevaar & R.Schoen, Global existence theorems for harmonic maps: finite rank spaces and an approach to rigidity for smooth actions, Preprint, 1998.

Consumer Warning:

Much of this material is new to the instructor, and the active help of anyone attending will be greatly appreciated.

This page is maintained by Robert Hardt ( email )
Last edited 08/22/00..